Rings whose nonsingular modules have projective covers
We determine rings $R$ with the property that all (finitely generated) nonsingular right $R$-modules have projective covers. These are just the rings with $t$-supplemented (finitely generated) free right modules. Hence, they are called right (finitely) $\Sigma -t$-supplemented. It is also shown that...
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| author | Asgari, Sh. Haghany, A. Асгарі, Ш. Хагані, А. |
| author_facet | Asgari, Sh. Haghany, A. Асгарі, Ш. Хагані, А. |
| author_sort | Asgari, Sh. |
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| description | We determine rings $R$ with the property that all (finitely generated) nonsingular right $R$-modules have projective covers.
These are just the rings with $t$-supplemented (finitely generated) free right modules. Hence, they are called right (finitely) $\Sigma -t$-supplemented. It is also shown that a ring $R$ for which every cyclic nonsingular right $R$-module has a projective cover is exactly a right $t$-supplemented ring. It is proved that, for a continuous ring $R$, the property of right $\Sigma -t$-supplementedness is equivalent to the semisimplicity of $R/Z_2(R_R)$, while the property of being right finitely $\Sigma -t$-supplemented is equivalent to the right self-injectivity of $R/Z_2(R_R)$. Moreover, for a von Neumann regular ring $R/Z_2(R_R)$, the properties of being right $\Sigma -t$-supplemented, right finitely \Sigma -t-supplemented, and right t-supplemented are equivalent to the semisimplicity, right self-injectivity, and right continuity of $R/Z_2(R_R)$, respectively. |
| first_indexed | 2026-03-24T02:13:12Z |
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| fulltext |
UDC 512.5
Sh. Asgari* (Univ. Isfahan, and School Math., Inst. Res. Fundam. Sci. (IPM), Tehran, Iran),
A. Haghany (Isfahan Univ. Technology, Iran)
RINGS WHOSE NONSINGULAR MODULES HAVE PROJECTIVE COVERS
КIЛЬЦЯ, ДЛЯ ЯКИХ НЕСИНГУЛЯРНI МОДУЛI МАЮТЬ
ПРОЕКТИВНI ПОКРИТТЯ
We determine rings R with the property that all (finitely generated) nonsingular right R-modules have projective covers.
These are just the rings with t-supplemented (finitely generated) free right modules. Hence, they are called right (finitely)
\Sigma -t-supplemented. It is also shown that a ring R for which every cyclic nonsingular right R-module has a projective cover
is exactly a right t-supplemented ring. It is proved that, for a continuous ring R, the property of right \Sigma -t-supplementedness
is equivalent to the semisimplicity of R/Z2(RR), while the property of being right finitely \Sigma -t-supplemented is equivalent
to the right self-injectivity of R/Z2(RR). Moreover, for a von Neumann regular ring R, the properties of being right
\Sigma -t-supplemented, right finitely \Sigma -t-supplemented, and right t-supplemented are equivalent to the semisimplicity, right
self-injectivity, and right continuity of R, respectively.
Визначено кiльця R з тiєю властивiстю, що всi (скiнченнопородженi) несингулярнi правi R-модулi мають проективнi
покриття. Це є саме кiльця з t-доповненими (скiнченнопородженими) вiльними правими модулями. Таким чином,
вони називаються правими (скiнченно) \Sigma -t-доповненими. Також показано, що кiльце R, для якого кожний циклiчний
несингулярний правий R-модуль має проективне покриття, є в точностi правим t-доповненим кiльцем. Доведено,
що для скiнченного кiльця R властивiсть правої \Sigma -t-доповненостi еквiвалентна напiвпростотi R/Z2(RR), а власти-
вiсть правої скiнченної \Sigma -t-доповненостi — правiй самоiн’єктивностi R/Z2(RR). Крiм того, для регулярного кiльця
фон Ноймана R властивостi правої \Sigma -t-доповненостi, правої скiнченної \Sigma -t-доповненостi та правої t-доповненостi
еквiвалентнi вiдповiдно напiвпростотi, правiй самоiн’єктивностi та правiй неперервностi R.
1. Introduction. Let R be a ring and \scrC be a class of right R-modules. For some special classes \scrC ,
the property of having a projective cover for each element of \scrC characterizes R. In [5], Bass studied
the rings R for which every element of \scrC has a projective cover, when \scrC is the class of all right
R-modules (resp., cyclic right R-modules). He called such rings right perfect rings (resp., semiperfect
rings). An excellent reference for a thorough study of these rings and their applications is [14]. When
\scrC is the class of semisimple right R-modules, each element of \scrC has a projective cover, if and only
if, R is right perfect; see [21] (43.9) and [17] (Theorem B.38). If \scrC is either the class of finitely
generated R-modules or the class of simple R-modules, then each element of \scrC has a projective cover,
if and only if, R is semiperfect [21] (42.6); and by [7] (Proposition 2.6), these are equivalent to R
being lifting. In [4], Azumaya called a ring R F-semiperfect if R/\mathrm{R}\mathrm{a}\mathrm{d}(R) is von Neumann regular
and idempotents can be lifted modulo \mathrm{R}\mathrm{a}\mathrm{d}(R). F-semiperfect rings are also known as semiregular
rings. If \scrC is the class of all factor modules R/I where I is a principal (finitely generated) right
ideal of R, then each element of \scrC has a projective cover, if and only if, R is semiregular; see [4]
(Proposition 1.7) and [17] (Theorem B.44). Moreover, when \scrC is the class of all singular right
R-modules, Guo in [12] showed that every element of \scrC has a projective cover, if and only if, R
is right perfect. So a natural question is: When \scrC is the class of all nonsingular right R-modules,
* The research of the first author was in part supported by a grant from IPM (№ 93160068).
c\bigcirc SH. ASGARI, A. HAGHANY, 2016
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1 3
4 SH. ASGARI, A. HAGHANY
which rings are determined by having the property that each element of \scrC has a projective cover?
We are more interested in characterizing such rings in a way similar to the characterization in [2] of
rings whose nonsingular modules are projective. We approach this by restricting the \oplus -supplemented
property to the t-closed submodules of projective modules, which we define in the following.
Throughout the paper, rings will have a nonzero identity element and modules will be unitary right
modules. Recall that a submodule K of a module M is called a supplement or ‘addition complement’
of a submodule A if K is minimal with respect to the property that A + K = M. Indeed, K is a
supplement of A, if and only if, A + K = M and A \cap K \ll K (the notation \ll denotes a small
submodule). A module M is called supplemented if any submodule A of M has a supplement in
M, and is called \oplus -supplemented if any submodule A of M has a supplement in M which is a
direct summand. For a projective module, the properties of supplemented and \oplus -supplemented are
equivalent. A t-closed submodule C of a module M (denoted by C \leq tc M ) is introduced in [2] as a
closed submodule of M which contains Z2(M). A module M is called t-extending if every t-closed
submodule of M is a direct summand, and a ring R is called right \Sigma -t-extending if all free right
R-modules are t-extending.
We say that a projective module P is t-supplemented if every t-closed submodule C of P has a
supplement in P which is a direct summand. In Section 2 we deal with t-supplemented projective
modules. Projective modules which are either t-extending or supplemented are t-supplemented. So
right extending rings and right lifting rings are right t-supplemented. It will be shown that projective
modules admit many characterizations for being t-supplemented (Theorem 2.1). The properties of
t-supplemented and t-extending coincide for projective modules with zero radical (Proposition 2.2),
and for projective modules over right continuous rings (Corollary 2.5).
In Section 3 we will prove that rings for which nonsingular modules have projective covers
are precisely rings whose all free modules are t-supplemented, called right \Sigma -t-supplemented rings
(Theorem 3.1). Following [3], a ring R is called right t-semisimple if R/Z2(RR) is semisimple. In
fact, R is right t-semisimple, if and only if, every nonsingular R-module is injective, if and only if,
every nonsingular R-module is semisimple. For rings we have
right t-semisimple \Rightarrow right \Sigma -t-extending \Rightarrow right \Sigma -t-supplemented
but none of these implications is reversible. The above properties coincide for a ring R such that
\mathrm{R}\mathrm{a}\mathrm{d}(R) \leq Z2(RR) (Proposition 3.3). In particular, for right continuous rings and rings with zero
radical, the properties of right \Sigma -t-supplemented, right \Sigma -t-extending, and right t-semisimple are
equivalent (Corollary 3.4). A right self-injective right \Sigma -t-supplemented ring R such that Z2(RR)
is either Noetherian or Artinian is exactly a quasi-Frobenius ring (Corollary 3.5). In the sequel, we
will see that rings whose finitely generated nonsingular modules have projective covers are exactly
right finitely \Sigma -t-supplemented rings (that is, all finitely generated free modules are t-supplemented).
Moreover, every nonsingular cyclic R-module has a projective cover, if and only if, R is right t-
supplemented. A right continuous ring R is right finitely \Sigma -t-supplemented if and only if R/Z2(RR)
is a right self-injective ring (Theorem 3.2). For a von Neumann regular ring R we obtain that: R
is right \Sigma -t-supplemented if and only if R is semisimple (Corollary 3.6); R is right finitely \Sigma -t-
supplemented if and only if R is right self-injective (Corollary 3.7); R is right t-supplemented if
and only if R is right continuous (Proposition 3.5). Finally, it is shown that the classes of right
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
RINGS WHOSE NONSINGULAR MODULES HAVE PROJECTIVE COVERS 5
\Sigma -t-supplemented rings, right finitely \Sigma -t-supplemented rings and right t-supplemented rings are
different (Example 3.3).
2. Projective modules with \bfitt -supplemented property. By considering the \oplus -supplemented
property to the t-closed submodules of a projective module we define the following notion.
Definition 2.1. We say that a projective module P is t-supplemented if any t-closed submodule
C of P has a supplement in P which is a direct summand, i.e., there exists a direct summand K of
P such that P = C +K and C \cap K \ll K. A ring R is called right t-supplemented if the module
RR is t-supplemented.
Projective modules which are either \oplus -supplemented or t-extending are t-supplemented. Hence
semiperfect rings, right lifting rings, right extending rings, and right Z2-torsion rings
\bigl(
that is,
Z2(RR) = R
\bigr)
are right t-supplemented.
The ring of integers \BbbZ is extending, and so it is t-supplemented, yet it is not \oplus -supplemented.
Hence the properties of \oplus -supplemented and t-supplemented are different for a projective module.
Moreover, the property of t-supplemented dose not coincide with the property of t-extending, as the
next result shows.
Proposition 2.1. Let R be a right perfect right nonsingular ring which is not right Artinian.
Then there exists a projective t-supplemented R-module P which is not t-extending.
Proof. Since R is right perfect, every projective R-module is supplemented. Thus by [13]
(Lemma 1.2), every projective R-module is \oplus -supplemented, and so it is t-supplemented. However,
not every projective R-module can be extending, for otherwise, R would be right \Sigma -t-extending by
[2] (Theorem 3.12(6)), hence right Artinian by [10]
\bigl(
12.21((a) \leftrightarrow (b))
\bigr)
.
In the following we give examples of rings which satisfy the conditions of Proposition 2.1.
Example 2.1. Let D be a division ring and \Lambda be an infinite set. Consider the upper triangular
matrix ring R =
\biggl(
D
\bigoplus
\Lambda D
0 D
\biggr)
. Clearly \mathrm{R}\mathrm{a}\mathrm{d}(R) =
\biggl(
0
\bigoplus
\Lambda D
0 0
\biggr)
. So R/\mathrm{R}\mathrm{a}\mathrm{d}(R) is semisimple
and \mathrm{R}\mathrm{a}\mathrm{d}(R) is nilpotent. Hence R is right perfect. Moreover, it is easy to see that R is right
nonsingular but not right Artinian.
Similarly the ring R =
\biggl(
D
\prod
\Lambda D
0 D
\biggr)
satisfies the conditions of Proposition 2.1.
The next result gives several equivalent conditions for a t-supplemented projective module.
Theorem 2.1. Let P be a projective module. The following statements are equivalent:
(1) P is t-supplemented.
(2) P/C has a projective cover for every t-closed submodule C of P.
(3) Every t-closed submodule C of P has a supplement which is projective.
(4) Every t-closed submodule C of P has a supplement which has a projective cover.
(5) Every t-closed submodule C of P has a supplement which has also a supplement.
(6) For every t-closed submodule C of P, there is a decomposition P = A\oplus K such that A \leq C
and C \cap K \ll K.
Proof. (1) \Rightarrow (6). Let C be a t-closed submodule of P. By hypothesis there exists a direct
summand K of P such that P = C +K and C \cap K \ll K. Assume that \pi : P \rightarrow P/C is the natural
epimorphism and f : K \rightarrow P/C is the small epimorphism with \mathrm{k}\mathrm{e}\mathrm{r}(f) = C\cap K. Since P is projective
we conclude that there exists a homomorphism g : P \rightarrow K such that fg = \pi . Hence fg(P ) = f(K),
and so g(P ) + (C \cap K) = K. Thus g(P ) = K, and g is an epimorphism. Since K is projective we
conclude that there exists a homomorphism h : K \rightarrow P such that gh = 1K . Thus P = \mathrm{k}\mathrm{e}\mathrm{r}(g)\oplus h(K)
and clearly \mathrm{k}\mathrm{e}\mathrm{r}(g) \leq C. If we show that C \cap h(K) \ll h(K), then P = \mathrm{k}\mathrm{e}\mathrm{r}(g)\oplus h(K) is the desired
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
6 SH. ASGARI, A. HAGHANY
decomposition of P. Let C\cap h(K) +X = h(K). So g(C\cap h(K))+g(X) = K. However, g(C) \leq C
hence (C \cap K) + g(X) = K, and so g(X) = K since C \cap K \ll K. Thus for each k \in K there
exists x \in X such that g(x) = k = gh(k). This implies that x - h(k) \in \mathrm{k}\mathrm{e}\mathrm{r}(g) \cap h(K) = 0 and so
X = h(K).
The implication (6) \Rightarrow (1) is clear, and the equivalences of (1) \leftrightarrow (2) \leftrightarrow (3) \leftrightarrow (4) \leftrightarrow (5) follow
from [4] (Proposition 1.4) and [22] (Proposition 2.1).
Corollary 2.1. The following statements are equivalent for a projective module P :
(1) P is t-supplemented.
(2) Every t-closed submodule C of P has a supplement K such that C \cap K is a direct summand
of C.
(3) For every t-closed submodule C of P, there exist a direct summand A of P and a small
submodule B of P such that C = A\oplus B.
(4) For every t-closed submodule C of P, there exists a direct summand A of P such that A \leq C
and C/A \ll P/A.
(5) For every t-closed submodule C of P, there exists an idempotent e \in \mathrm{E}\mathrm{n}\mathrm{d}(P ) such that
eP \leq C and (1 - e)C \ll (1 - e)P.
Proof. This follows from Theorem 2.1(6) and [9] (22.1).
Corollary 2.2. If P is a projective t-supplemented module, then so is every direct summand of P.
Proof. Let P = P1 \oplus P2 and C1 \leq tc P1. Clearly P/(C1 \oplus P2) \sim = P1/C1. Thus by [2]
(Proposition 2.6(6)), C1 \oplus P2 \leq tc P and so by Theorem 2.1(2), P/(C1 \oplus P2) hence P1/C1 has a
projective cover. Therefore P1 is t-supplemented by Theorem 2.1(2).
Let U and N be submodules of a module M. It is said that U respects N if there exists a
decomposition M = A \oplus K such that A \leq N and N \cap K \leq U. In [19] it is shown that R
is a semiperfect ring, if and only if, \mathrm{R}\mathrm{a}\mathrm{d}(R) respects every right ideal of R. Moreover, by [19]
(Theorem 28) and [17] (Lemma B.40), R is a semiregular ring, if and only if, \mathrm{R}\mathrm{a}\mathrm{d}(R) respects
every finitely generated (principal) right ideal of R. The next result shows that a ring R is right
t-supplemented, if and only if, \mathrm{R}\mathrm{a}\mathrm{d}(R) respects every t-closed right ideal of R.
Corollary 2.3. Let P be a projective module such that \mathrm{R}\mathrm{a}\mathrm{d}(P ) \ll P. The following statements
are equivalent:
(1) P is t-supplemented.
(2) \mathrm{R}\mathrm{a}\mathrm{d}(P ) respects every t-closed submodule of P.
Proof. The implication (1) \Rightarrow (2) is clear by Theorem 2.1(6), and the implication (2) \Rightarrow (1)
follows from Corollary 2.1(3) and [20] (Lemma 3.1).
The next result shows that the properties of t-extending and t-supplemented coincide for a
projective module with zero radical.
Proposition 2.2. If P is a t-supplemented projective module, then P/\mathrm{R}\mathrm{a}\mathrm{d}(P ) is t-extending.
Proof. Let C/\mathrm{R}\mathrm{a}\mathrm{d}(P ) \leq tc P/\mathrm{R}\mathrm{a}\mathrm{d}(P ). Then C \leq tc P and so there exists a direct summand K
of P such that P = C +K and C \cap K \ll K. Therefore C \cap K \leq \mathrm{R}\mathrm{a}\mathrm{d}(P ) and
P/\mathrm{R}\mathrm{a}\mathrm{d}(P ) = C/\mathrm{R}\mathrm{a}\mathrm{d}(P )\oplus (K +\mathrm{R}\mathrm{a}\mathrm{d}(P ))/\mathrm{R}\mathrm{a}\mathrm{d}(P ).
Hence P/\mathrm{R}\mathrm{a}\mathrm{d}(P ) is t-extending.
Proposition 2.3. Let R be either a right Noetherian ring or an exchange ring for which \mathrm{R}\mathrm{a}\mathrm{d}(R)
is Z2-torsion. A finitely generated projective R-module P is t-supplemented if and only if P/\mathrm{R}\mathrm{a}\mathrm{d}(P )
is t-extending.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
RINGS WHOSE NONSINGULAR MODULES HAVE PROJECTIVE COVERS 7
Proof. (\Rightarrow ). This follows from Proposition 2.2.
(\Leftarrow ). Let C be a t-closed submodule of P and P denote the factor module P/\mathrm{R}\mathrm{a}\mathrm{d}(P ). Since
\mathrm{R}\mathrm{a}\mathrm{d}(R) is Z2-torsion we conclude that \mathrm{R}\mathrm{a}\mathrm{d}(P ) is Z2-torsion. Thus by [2] (Lemma 2.5(1)),
\mathrm{R}\mathrm{a}\mathrm{d}(P ) \leq C, and so C is a t-closed R-submodule of P by [2] (Proposition 2.6(6)). The t-
extending property of P implies that there exists a decomposition P = C \oplus L. Hence P = C + L,
and C \cap L = \mathrm{R}\mathrm{a}\mathrm{d}(P ) \ll P. Let R be right Noetherian. Then P is Noetherian, and so C is finitely
generated. Therefore C \cap L = \mathrm{R}\mathrm{a}\mathrm{d}(C) \ll C. Similarly, C \cap L \ll L. Thus by [21] (41.14(2)),
P = C \oplus L, and so P is t-supplemented. Now assume that R is an exchange ring. By [9] (11.9),
P has the exchange property. Thus by [6] (Theorem 3), there exist submodules B \leq C and K \leq L
such that P = B \oplus K. Therefore P = C +K and C \cap K \leq C \cap L \ll P. This shows that K is a
direct summand of P which is a weak supplement, hence a supplement of C.
The following results give more relations between the properties of t-supplemented and t-
extending for a projective module.
Proposition 2.4. The following statements are equivalent for a projective module P :
(1) P is t-extending.
(2) P is t-supplemented and every t-closed submodule of P is a supplement.
If \mathrm{R}\mathrm{a}\mathrm{d}(P ) \ll P, then the above statements are equivalent to
(3) P is t-supplemented and C \cap \mathrm{R}\mathrm{a}\mathrm{d}(P ) = \mathrm{R}\mathrm{a}\mathrm{d}(C) \ll C for every t-closed submodule C.
Proof. (1) \Rightarrow (2). This is clear by the property of t-extending.
(2) \Rightarrow (1). Let C be a t-closed submodule of P. There exists a direct summand K of P such that
P = C +K and C \cap K \ll K. By [9] (20.4(9)), C is a supplement of K. Therefore by [9] (20.9),
P = C \oplus K.
(1) \Rightarrow (3). Since each t-closed submodule C is a direct summand, C \cap \mathrm{R}\mathrm{a}\mathrm{d}(P ) = \mathrm{R}\mathrm{a}\mathrm{d}(C) by
[9] (20.4(7)). On the other hand, \mathrm{R}\mathrm{a}\mathrm{d}(P ) \ll P implies that \mathrm{R}\mathrm{a}\mathrm{d}(C) \ll P, hence \mathrm{R}\mathrm{a}\mathrm{d}(C) \ll C.
(3) \Rightarrow (1). Let C be a t-closed submodule of P. There exists a direct summand K of P such that
P = C +K and C \cap K \ll K. Thus C \cap K \leq \mathrm{R}\mathrm{a}\mathrm{d}(K) and by hypothesis C \cap K \leq \mathrm{R}\mathrm{a}\mathrm{d}(C). Now
consider the epimorphism f : C\oplus K \rightarrow P which is defined by f(c, k) = c+k. Since P is projective
we conclude that f splits and so \mathrm{k}\mathrm{e}\mathrm{r}(f) is a direct summand of C \oplus K. Clearly, \mathrm{k}\mathrm{e}\mathrm{r}(f) = \{ (x, - x) :
x \in C \cap K\} \leq \mathrm{R}\mathrm{a}\mathrm{d}(C) \oplus \mathrm{R}\mathrm{a}\mathrm{d}(K) = \mathrm{R}\mathrm{a}\mathrm{d}(C \oplus K). But K is a direct summand of P and
\mathrm{R}\mathrm{a}\mathrm{d}(P ) \ll P, hence \mathrm{R}\mathrm{a}\mathrm{d}(K) \ll K. So \mathrm{R}\mathrm{a}\mathrm{d}(C \oplus K) \ll C \oplus K. Thus \mathrm{k}\mathrm{e}\mathrm{r}(f) \ll C \oplus K, and so
\mathrm{k}\mathrm{e}\mathrm{r}(f) = 0. Hence P \sim = C \oplus K which implies that C is a direct summand of P.
Corollary 2.4. Let P be a projective module such that \mathrm{R}\mathrm{a}\mathrm{d}(P ) is Z2-torsion. The following
statements are equivalent:
(1) P is t-extending.
(2) P is t-supplemented and Z2(P ) is a supplement.
(3) P is t-supplemented and Z2(P ) is a direct summand of P.
Proof. (1) \Rightarrow (3) \Rightarrow (2). These are obvious.
(2) \Rightarrow (1). Since \mathrm{R}\mathrm{a}\mathrm{d}(P ) is Z2-torsion we conclude that P/Z2(P ) is a homomorphic image of
P/\mathrm{R}\mathrm{a}\mathrm{d}(P ). Hence P/Z2(P ) is t-extending by Proposition 2.2. Let C be a t-closed submodule of
P. Clearly, C/Z2(P ) is t-closed in P/Z2(P ), and so it is a direct summand of P/Z2(P ). Hence by
[9] (20.5(2)), C is a supplement in M. Thus P is t-extending by Proposition 2.4.
Corollary 2.5. Let R be a right continuous ring. Then the properties of t-supplemented and
t-extending coincide for a projective R-module P.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
8 SH. ASGARI, A. HAGHANY
Proof. Since Z2(RR) is a direct summand of R we conclude that Z2(F ) is a direct summand
of F, for every free R-module F. Thus Z2(P ) is a direct summand of P. On the other hand,
\mathrm{R}\mathrm{a}\mathrm{d}(R) = Z(RR) is Z2-torsion. Hence \mathrm{R}\mathrm{a}\mathrm{d}(F ) is Z2-torsion for every free R-module F, and
this implies that \mathrm{R}\mathrm{a}\mathrm{d}(P ) is Z2-torsion. So by Corollary 2.4, the properties of t-supplemented and
t-extending are equivalent for P.
3. Right (finitely) \bfSigma -\bfitt -supplemented rings. In this section, we show that a ring R whose all
(resp., all finitely generated) nonsingular R-modules have projective covers is precisely a ring R for
which all (resp., all finitely generated) free R-modules are t-supplemented. Note that a direct sum
of t-supplemented free modules need not be t-supplemented. For example, if R = \BbbZ [x] then by [8]
(Example 2.4), R is an extending R-module but R \oplus R is not so. For, R being right nonsingular,
the properties of extending and t-extending are the same and since \mathrm{R}\mathrm{a}\mathrm{d}(R) = 0, the notions of
t-extending and t-supplemented are equivalent by Proposition 2.2. Hence for this ring, a direct sum
of t-supplemented free modules need not be t-supplemented.
Definition 3.1. We say that a ring R is right \Sigma -t-supplemented if every free R-module is t-
supplemented.
Recall from [2] that a ring R is right \Sigma -t-extending if every free R-module is t-extending. Clearly
right \Sigma -t-extending rings are right \Sigma -t-supplemented. The next example show that the class of right
\Sigma -t-supplemented rings properly contains the class of right \Sigma -t-extending rings.
Example 3.1. Let R =
\biggl(
\BbbQ \BbbR
0 \BbbQ
\biggr)
. Since \mathrm{R}\mathrm{a}\mathrm{d}(R) =
\biggl(
0 \BbbR
0 0
\biggr)
is nilpotent and R/\mathrm{R}\mathrm{a}\mathrm{d}(R) is
semisimple, we conclude that R is right perfect and so it is right \Sigma -t-supplemented. However it is
easy to see that R is right nonsingular, hence if it were right \Sigma -t-extending then R would be right
Artinian by [10] (12.21(b)), which is not. Hence R is not right \Sigma -t-extending.
The following result gives some equivalent conditions for a ring R with the property that all
nonsingular R-modules have projective covers. The equivalence (1) \leftrightarrow (4) is in contrast with [2]\bigl(
Theorem 3.12((1) \leftrightarrow (2))
\bigr)
. For brevity let us say that a module M satisfies the property \scrP if every
t-closed submodule of M has a supplement in M which has a projective cover.
Theorem 3.1. The following statements are equivalent for a ring R :
(1) R is right \Sigma -t-supplemented.
(2) Every projective R-module is t-supplemented.
(3) Every R-module M satisfies the property \scrP .
(4) Every nonsingular R-module has a projective cover.
Proof. (1) \Rightarrow (3). Let M be an R-module. There exists a free R-module F such that M \sim = F/L
for some submodule L of F. Then it suffices to show that every t-closed submodule of F/L has a
supplement in F/L which has a projective cover. Assume that C/L is a t-closed submodule of F/L.
Then C is t-closed in F by [2] (Proposition 2.6(6)). By Theorem 2.1(4), there exists a submodule K
of F such that K has a projective cover, F = C+K and C\cap K \ll K. Thus F/L = C/L+(K+L)/L
and C/L \cap (K + L)/L \ll (K + L)/L. Moreover, C \cap K \ll K implies that L \cap K \ll K and
so if f : P \rightarrow K is a projective cover, then \pi f : P \rightarrow (K + L)/L is a projective cover where \pi :
K \rightarrow (K + L)/L is the canonical projection. Hence (K + L)/L is a supplement of C/L which has
a projective cover.
(3) \Rightarrow (4). Let M be a nonsingular R-module. By [2] (Proposition 2.6(6)), the zero submodule
is t-closed in M. Thus by hypothesis, M has a projective cover.
(4) \Rightarrow (2). Let P be a projective R-module and C be a t-closed submodule of P. Then P/C is
nonsingular and so it has a projective cover. Hence P is t-supplemented by Theorem 2.1(2).
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RINGS WHOSE NONSINGULAR MODULES HAVE PROJECTIVE COVERS 9
(2) \Rightarrow (1). This is clear.
Corollary 3.1. If R is a right \Sigma -t-supplemented ring, then so is R/Z2(RR).
Proof. Let R = R/Z2(RR) and M be a nonsingular R-module. If m \in Z(MR), then there
exists a essential right ideal I of R such that mI = 0. By [2] (Proposition 2.2(2)), I is an essential
right ideal of R. So mI = 0 and the nonsingular property of MR imply that m = 0. Hence MR is
nonsingular. Thus by Theorem 3.1(4), MR has a projective cover. So by [16] (Lemma 24.15), MR
has a projective cover. Hence R is a right \Sigma -t-supplemented ring by Theorem 3.1(4).
Corollary 3.2. Every right perfect ring is right \Sigma -t-supplemented.
Proof. This is clear by Theorem 3.1(4).
The next example show that the class of right \Sigma -t-supplemented rings properly contains the class
of right perfect rings. The following lemma is helpful.
Lemma 3.1. If R =
\prod
\Lambda
R\lambda where each R\lambda is a ring, then Z(RR) =
\prod
\Lambda
Z((R\lambda )R\lambda
) and
Z2(RR) =
\prod
\Lambda
Z2 ((R\lambda )R\lambda
) .
Proof. It is easy to see that a right ideal I of R is essential if and only if I contains
\bigoplus
\Lambda I\lambda
where I\lambda is an essential right ideal in (R\lambda )R\lambda
. This implies that Z(RR) =
\prod
\Lambda
Z((R\lambda )R\lambda
) and so
Z2(RR) =
\prod
\Lambda
Z2((R\lambda )R\lambda
).
Example 3.2. Let R =
\prod
\Lambda
\BbbZ /4\BbbZ , where \Lambda is an infinite set. Since Z2(\BbbZ /4\BbbZ ) = \BbbZ /4\BbbZ ,
Lemma 3.1 implies that Z2(RR) = R. However MZ2(RR) \leq Z2(M), for every R-module M.
Therefore Z2(M) = M and so M is t-extending. Thus R is right \Sigma -t-extending hence it is right
\Sigma -t-supplemented. However \mathrm{R}\mathrm{a}\mathrm{d}(R) =
\prod
\Lambda
\mathrm{R}\mathrm{a}\mathrm{d}(\BbbZ /4\BbbZ ) =
\prod
\Lambda
2\BbbZ /4\BbbZ and so R/\mathrm{R}\mathrm{a}\mathrm{d}(R) \sim =
\sim =
\prod
\Lambda
\BbbZ /2\BbbZ is not semisimple. Thus R is not a right perfect ring.
Proposition 3.1. LetR be a right \Sigma -t-supplemented ring. IfZ2(RR) is semiprime, thenR/Z2(RR)
is a right hereditary ring.
Proof. Since R/Z2(RR) is a nonsingular R-module we conclude that R/Z2(RR) is a right
nonsingular ring. Hence every free R/Z2(RR)-module is nonsingular and so is every submodule of
a projective R/Z2(RR)-module. But R/Z2(RR) is a right \Sigma -t-supplemented ring by Corollary 3.1.
Therefore by Theorem 3.1(4), submodules of projective R/Z2(RR)-modules have projective covers.
Thus by [11] (Corollary 1.6), R/Z2(RR) is a right hereditary ring.
Corollary 3.3. Let R be a right \Sigma -t-supplemented right t-extending ring. If Z2(RR) is semiprime,
then R/Z2(RR) is a right Noetherian ring.
Proof. By Proposition 3.1, R/Z2(RR) is right hereditary. Moreover, R/Z2(RR) is an extending
R-module by [2] (Theorem 2.11(3)). Hence R/Z2(RR) is a right extending ring, and so it is a right
Noetherian ring by [10] (Corollary 10.6(1)).
Proposition 3.2. Let R be a right \Sigma -t-supplemented ring. Then R is a right max ring with the
zero radical if and only if R is a right V-ring.
Proof. Let R be a max ring with the zero radical and M be an R-module. By Theorem 3.1(3),
Z2(M/\mathrm{R}\mathrm{a}\mathrm{d}(M)) has a supplement K/\mathrm{R}\mathrm{a}\mathrm{d}(M) which has a projective cover. Since \mathrm{R}\mathrm{a}\mathrm{d}(R) = 0
we conclude that K/\mathrm{R}\mathrm{a}\mathrm{d}(M) is projective, and so \mathrm{R}\mathrm{a}\mathrm{d}(K) = \mathrm{R}\mathrm{a}\mathrm{d}(M) is a direct summand of K.
But R is a max ring, and so \mathrm{R}\mathrm{a}\mathrm{d}(K) \ll K. Hence \mathrm{R}\mathrm{a}\mathrm{d}(M) = 0. This implies that R is a right
V -ring; see [21] (23.1). The converse is clear.
Recall from [3] that R is a right t-semisimple ring if R/Z2(RR) is semisimple. There, it was
shown that R is right t-semisimple, if and only if, every nonsingular R-module is injective, if and
only if, every nonsingular R-module is semisimple, if and only if, every nonsingular right ideal of R
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
10 SH. ASGARI, A. HAGHANY
is a direct summand. The following result shows that for a ring R such that \mathrm{R}\mathrm{a}\mathrm{d}(R) \leq Z2(RR), the
properties of right \Sigma -t-supplemented, right \Sigma -t-extending, and right t-semisimple are equivalent.
Proposition 3.3. The following statements are equivalent for a ring R :
(1) R is right \Sigma -t-supplemented and \mathrm{R}\mathrm{a}\mathrm{d}(R) is Z2-torsion.
(2) R is right \Sigma -t-extending and \mathrm{R}\mathrm{a}\mathrm{d}(R) is Z2-torsion.
(3) R is right t-semisimple.
Proof. (1) \Rightarrow (3). Let R(\Lambda ) be a free R-module. By hypothesis, R(\Lambda ) is t-supplemented and
so R(\Lambda )/\mathrm{R}\mathrm{a}\mathrm{d}(R(\Lambda )) is t-extending by Proposition 2.2. Since \mathrm{R}\mathrm{a}\mathrm{d}(RR) is Z2-torsion we conclude
that \mathrm{R}\mathrm{a}\mathrm{d}(R(\Lambda )) is Z2-torsion. Thus by [2] (Proposition 2.14(1)), [R/Z2(RR)]
(\Lambda ) \sim = R(\Lambda )/Z2(R
(\Lambda ))
is t-extending. But [R/Z2(RR)]
(\Lambda ) is nonsingular, and so [R/Z2(RR)]
(\Lambda ) is extending. This implies
that R/Z2(RR) is a right \Sigma -extending ring. On the other hand, R/Z2(RR) is a right nonsingular ring
since it is a nonsingular R-module. Hence R/Z2(RR) is right Artinian by [10] (12.21(b)). Therefore
R is right t-semisimple by [3] (Corollary 4.4(1)).
(3) \Rightarrow (2). This follows by [3] (Corollary 3.6 and Theorem 2.3(3)).
(2) \Rightarrow (1). This implication is obvious.
Corollary 3.4. For right continuous rings and rings with zero radical, the properties of right
\Sigma -t-supplemented, right \Sigma -t-extending, and right t-semisimple are equivalent.
Recall that a ring R is called quasi-Frobenius if R is right or left Artinian and right or left
self-injective ring (all cases are equivalent). It is well known that R is quasi-Frobenius if and only
if R is left and right self-injective and left or right perfect; see [17] (Theorem 6.39). The Faith
conjecture states that every left or right perfect, right self-injective ring R is quasi-Frobenius. This
conjecture remains open, but imposing extra condition(s) on R ensures that R is quasi-Frobenius; see
[17]. The next result, in particular, shows that a right self-injective right perfect ring with Noetherian
or Artinian second singular ideal is exactly a quasi-Frobenius ring.
Corollary 3.5. The following statements are equivalent:
(1) R is a right self-injective right \Sigma -t-supplemented ring such that Z2(RR) is Noetherian.
(2) R is a right self-injective right \Sigma -t-supplemented ring such that Z2(RR) is Artinian.
(3) R is a quasi-Frobenius ring.
Proof. (1) \Rightarrow (3). By Corollary 3.4, R/Z2(RR) is semisimple. Therefore R/Z2(RR) is a
Noetherian R-module, and so by hypothesis, R is right Noetherian. Thus R is quasi-Frobenius.
Similarly, the implication (2) \Rightarrow (3) can be proved, and clearly, (3) \Rightarrow (1), (2).
Proposition 3.4. The following statements are equivalent for a ring R :
(1) R is right \Sigma -t-supplemented and \mathrm{R}\mathrm{a}\mathrm{d}(R) = Z2(RR).
(2) R is right \Sigma -t-extending and \mathrm{R}\mathrm{a}\mathrm{d}(R) = Z2(RR).
(3) R is semisimple.
Proof. (1) \Rightarrow (3). By Proposition 3.3, R is right t-semisimple. So Z2(RR) is a direct summand
of R by [3] (Theorem 2.3(3)). However, Z2(RR) = \mathrm{R}\mathrm{a}\mathrm{d}(R) implies that Z2(RR) = 0. Hence R is
semisimple.
(3) \Rightarrow (2) \Rightarrow (1). These implications are clear.
Corollary 3.6. Let R be a von Neumann regular ring. Then R is right \Sigma -t-supplemented if and
only if R is semisimple.
Proof. This follows from Proposition 3.4.
In the following we consider rings for which every finitely generated (resp., cyclic) nonsingular
R-module has a projective cover. Let us call a ring R right finitely \Sigma -t-supplemented if every finitely
generated free R-module is t-supplemented.
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RINGS WHOSE NONSINGULAR MODULES HAVE PROJECTIVE COVERS 11
Remark 3.1. The proof of Theorem 3.1 shows that similar equivalent conditions hold for R if
in the statements we replace ‘right \Sigma -t-supplemented’ by ‘right finitely \Sigma -t-supplemented’, ‘right
\Sigma -t-extending’ by ‘right finitely \Sigma -t-extending’ (see [2], Remark 3.14), and assume that R-modules
under consideration are finitely generated. So a ring R for which every finitely generated nonsingular
R-module has a projective cover is precisely a right finitely \Sigma -t-supplemented ring. Thus every
semiperfect ring is right finitely \Sigma -t-supplemented. However the properties of right finitely \Sigma -t-
supplemented and semiperfect are not equivalent; for example, the ring R =
\prod
\Lambda
\BbbZ /4\BbbZ (for an
infinite set \Lambda ) is right finitely \Sigma -t-supplemented as shown in Example 3.2, but R/\mathrm{R}\mathrm{a}\mathrm{d}(R) is not
semisimple and so R is not semiperfect.
In Corollary 3.4, the property of right \Sigma -t-supplemented for right continuous rings and rings with
zero radicals is characterized. In the following, we determine when a right continuous ring is right
finitely \Sigma -t-supplemented.
Theorem 3.2. The following statements are equivalent for a right continuous ring R :
(1) R is right finitely \Sigma -t-supplemented.
(2) R is right finitely \Sigma -t-extending.
(3) R/Z2(RR) is a right self-injective ring.
(4) M/\mathrm{R}\mathrm{a}\mathrm{d}(M) is t-extending for every finitely generated (free, projective) R-module M.
Proof. The equivalences of (1), (2) follows from Corollary 2.5.
(2) \Rightarrow (3). Clearly, hypothesis implies that R\lambda /Z2((R\lambda )R\lambda
) is a right continuous right finitely \Sigma -
extending ring. So R\lambda /Z2((R\lambda )R\lambda
) is a right self-injective ring by [17] (Corollary 7.41((1) \leftrightarrow (3))).
Hence R/Z2(RR) is a right self-injective ring.
(3) \Rightarrow (2). Since R is right continuous, Z2(RR) is a direct summand of R. So R/Z2(RR) is a
projective R-module. Hence by [15] (Corollary 3.6A), R/Z2(RR) is an injective R-module. Thus R
is right finitely \Sigma -t-extending by [2] (Theorem 2.11(3)).
(1) \Rightarrow (4). Let M be a finitely generated R-module. There exists an epimorphism f :
F \rightarrow M for some finitely generated free R-module F. Clearly f(\mathrm{R}\mathrm{a}\mathrm{d}(F )) \leq \mathrm{R}\mathrm{a}\mathrm{d}(M) and so f :
F/\mathrm{R}\mathrm{a}\mathrm{d}(F ) \rightarrow M/\mathrm{R}\mathrm{a}\mathrm{d}(M) defined by f(x+\mathrm{R}\mathrm{a}\mathrm{d}(F )) = f(x)+\mathrm{R}\mathrm{a}\mathrm{d}(M), is an epimorphism. How-
ever, F/\mathrm{R}\mathrm{a}\mathrm{d}(F ) is t-extending by Proposition 2.2, and so M/\mathrm{R}\mathrm{a}\mathrm{d}(M) is t-extending by [2] (Propo-
sition 2.14(1)).
(4) \Rightarrow (1). Let F be a finitely generated free R-module. Since R is right continuous, R is an
exchange ring and \mathrm{R}\mathrm{a}\mathrm{d}(R) is Z2-torsion. Thus by Proposition 2.3, F is t-supplemented.
Corollary 3.7. A von Neumann regular ring R is right finitely \Sigma -t-supplemented if and only if it
is right self-injective.
Proof. (\Rightarrow ). By Proposition 2.2, every finitely generated free R-module is t-extending. So R
is right finitely \Sigma -t-extending. Hence R is right extending as it is right nonsingular. Since R is
von Neumann regular we conclude that R is right continuous. Thus by Theorem 3.2(3), R is right
self-injective.
(\Leftarrow ). Since every finitely generated free R-module is injective, we conclude that R is right finitely
\Sigma -t-extending. So it is right finitely \Sigma -t-supplemented.
By [10] (18.26), R is quasi-Frobenius, if and only if, R is left and right continuous and left and
right Artinian. There are examples of one-sided continuous left and right Artinian rings which are not
quasi-Frobenius; see [10] (Examples 18.27). In [1] and [18] one can find more conditions on a right
continuous ring to be quasi-Frobenius. The next result, in particular, shows that a right continuous
right Artinian ring with injective second singular ideal is exactly a quasi-Frobenius ring.
Corollary 3.8. The following statements are equivalent:
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
12 SH. ASGARI, A. HAGHANY
(1) R is a right continuous right finitely \Sigma -t-supplemented ring such that Z2(RR) is injective.
(2) R is right self-injective ring.
Proof. (1) \Rightarrow (2). As shown in the proof of Theorem 3.2 ((3) \Rightarrow (2)), R/Z2(RR) is an injective
R-module. Since Z2(RR) is a direct summand of R we conclude that R is right self-injective.
(2) \Rightarrow (1). This is obvious.
Remark 3.2. Recall that every cyclic R-module has a projective cover, if and only if, R is a
semiperfect ring, if and only if, R is right supplemented. By modifying the proof of Theorem 3.1,
similar equivalent conditions hold for R if in the statements we replace ‘right \Sigma -t-supplemented’
by ‘right t-supplemented’, ‘right \Sigma -t-extending’ by ‘right t-extending’, and assume that R-modules
under consideration are cyclic. Hence a ring R for which every nonsingular cyclic R-module has a
projective cover is exactly a right t-supplemented ring (which is characterized in Theorem 2.1 and
Corollary 2.3).
The next result is in contrast with Corollaries 3.6 and 3.7.
Proposition 3.5. A von Neumann regular ring R is right t-supplemented if and only if it is right
continuous.
Proof. Let R be right t-supplemented. By Proposition 2.2, R is right t-extending. So R is right
extending as it is right nonsingular. On the other hand, R has the C2 condition. Thus R is right
continuous. The converse implication is clear since every continuous module is t-extending by [2]
(Theorem 2.11(3)).
Finally we give examples showing that the classes of right \Sigma -t-supplemented rings, right finitely
\Sigma -t-supplemented rings and right t-supplemented rings are indeed different.
Example 3.3. (i) Let F be a field and F \prime be a proper subfield of F. Set S =
\prod
\BbbN
F and assume
that R is the subring of S consisting of all (an)\BbbN with an \in F \prime for all but a finite number of
elements n \geq 1. As shown in [10] (Examples 12.20(i)), R is a commutative von Neumann regular
ring which is extending but not finitely \Sigma -extending. Since every von Neumann regular ring is
nonsingular with zero Jacobson radical, Proposition 2.2 shows that R is t-supplemented but not
finitely \Sigma -t-supplemented.
(ii) Let F be a field and V be an infinite dimensional vector space over F. Then consider
the von Neumann regular ring R = \mathrm{E}\mathrm{n}\mathrm{d}(V ). As shown in [10] (Examples 12.20(ii)), R is right
finitely \Sigma -extending but not right \Sigma -extending. Again, Proposition 2.2 implies that R is right finitely
\Sigma -t-supplemented but not right \Sigma -t-supplemented.
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Received 21.10.13,
after revision — 06.03.15
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
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| resource_txt_mv | umjimathkievua/1d/20b250e8796436690ef5939a35f8c01d.pdf |
| spelling | umjimathkievua-article-18172019-12-05T09:28:56Z Rings whose nonsingular modules have projective covers Кiльця, для яких несингулярнi модулi мають проективнi покриття Asgari, Sh. Haghany, A. Асгарі, Ш. Хагані, А. We determine rings $R$ with the property that all (finitely generated) nonsingular right $R$-modules have projective covers. These are just the rings with $t$-supplemented (finitely generated) free right modules. Hence, they are called right (finitely) $\Sigma -t$-supplemented. It is also shown that a ring $R$ for which every cyclic nonsingular right $R$-module has a projective cover is exactly a right $t$-supplemented ring. It is proved that, for a continuous ring $R$, the property of right $\Sigma -t$-supplementedness is equivalent to the semisimplicity of $R/Z_2(R_R)$, while the property of being right finitely $\Sigma -t$-supplemented is equivalent to the right self-injectivity of $R/Z_2(R_R)$. Moreover, for a von Neumann regular ring $R/Z_2(R_R)$, the properties of being right $\Sigma -t$-supplemented, right finitely \Sigma -t-supplemented, and right t-supplemented are equivalent to the semisimplicity, right self-injectivity, and right continuity of $R/Z_2(R_R)$, respectively. Визначено кiльця $R$ з тiєю властивiстю, що всi (скiнченнопородженi) несингулярнi правi $R$-модулi мають проективнi покриття. Це є саме кiльця з $t$-доповненими (скiнченнопородженими) вiльними правими модулями. Таким чином, вони називаються правими (скiнченно) $\Sigma -t$-доповненими. Також показано, що кiльце $R$, для якого кожний циклiчний несингулярний правий $R$-модуль має проективне покриття, є в точностi правим $t$-доповненим кiльцем. Доведено, що для скiнченного кiльця $R$ властивiсть правої $\Sigma -t$-доповненостi еквiвалентна напiвпростотi $R/Z_2(R_R)$, а власти- вiсть правої скiнченної $\Sigma -t$-доповненостi — правiй самоiн’єктивностi $R/Z_2(R_R)$. Крiм того, для регулярного кiльця фон Ноймана $R$ властивостi правої $\Sigma -t$-доповненостi, правої скiнченної $\Sigma -t$-доповненостi та правої $t$-доповненостi еквiвалентнi вiдповiдно напiвпростотi, правiй самоiн’єктивностi та правiй неперервностi $R$. Institute of Mathematics, NAS of Ukraine 2016-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1817 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 1 (2016); 3-13 Український математичний журнал; Том 68 № 1 (2016); 3-13 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1817/799 Copyright (c) 2016 Asgari Sh.; Haghany A. |
| spellingShingle | Asgari, Sh. Haghany, A. Асгарі, Ш. Хагані, А. Rings whose nonsingular modules have projective covers |
| title | Rings whose nonsingular modules have projective covers |
| title_alt | Кiльця, для яких несингулярнi модулi мають проективнi покриття |
| title_full | Rings whose nonsingular modules have projective covers |
| title_fullStr | Rings whose nonsingular modules have projective covers |
| title_full_unstemmed | Rings whose nonsingular modules have projective covers |
| title_short | Rings whose nonsingular modules have projective covers |
| title_sort | rings whose nonsingular modules have projective covers |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1817 |
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