I-radicals and right perfect rings
We determine the rings for which every hereditary torsion theory is an S-torsion theory in the sense of Komarnitskiy. We show that such rings admit a primary decomposition. Komarnitskiy obtained this result in the special case of left duo rings.
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2007
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509439904186368 |
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| author | Rump, W. Рамп, У. |
| author_facet | Rump, W. Рамп, У. |
| author_sort | Rump, W. |
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| description | We determine the rings for which every hereditary torsion theory is an S-torsion theory in the sense of Komarnitskiy.
We show that such rings admit a primary decomposition. Komarnitskiy obtained this result in the special case of left duo rings. |
| first_indexed | 2026-03-24T02:41:08Z |
| format | Article |
| fulltext |
UDC 512.5
W. Rump (Inst. Algebra und Zahlentheorie, Univ. Stuttgart, Germany)
I-RADICALS AND RIGHT PERFECT RINGS
I-RADYKALY TA DOSKONALI SPRAVA KIL|CQ
We determine the rings for which every hereditary torsion theory is an S-torsion theory in the sense of
Komarnitskiy. We show that such rings admit a primary decomposition. Komarnitskiy obtained this
result in the special case of left duo rings.
Vyznaçeno kil\cq, dlq qkyx koΩna teoriq skrutu z uspadkuvannqm [ teori[g S-skrutu u sensi
Komarnyc\koho. Pokazano, wo taki kil\cq dopuskagt\ pervynnyj rozklad. Komarnyc\kyj
otrymav cej rezul\tat u çastynnomu vypadku livyx duo-kilec\.
The concept of I-radical (defined below) was introduced by O. Horbachuk (see [1]) and
further developed and applied in collaboration with Yu. Maturin [2 – 5]. Any ideal I
of a ring R gives rise to an I-radical, and the lattice of I-radicals is always distributive
[4]. It is natural to ask about the relationship between I-radicals and Gabriel
topologies, that is, left exact radicals of R. In [5], it is proved that a ring R with the
property
(P) Every left exact radical in R-Mod is an I-radical
is right perfect, while the converse does not hold in general [6]. Komarnitskiy [7]
proved the converse in case of a left duo ring R (see also [2]). He showed that such
rings R admit a primary decomposition [8].
In this note, we prove that a ring satisfies (P) if and only if it decomposes into
finitely many quasilocal right perfect rings. This shows that the rings with property (P)
coincide with the rings studied by M. Teply [9], i.e., those for which the global
dimension with respect to each hereditary torsion theory is zero.
Let R be a ring (associative with 1). The category of left (right) R-modules will
be denoted by R-Mod (resp. Mod-R ), and N ≤ M will indicate that N is a
submodule of M. Recall that a pair ( T, F ) of full subcategories of R-Mod is said to
be a torsion theory [8] if T and F are maximal with respect to HomR T F( , ) = 0 for
all T ∈ T and F ∈ F . The torsion class T of ( T, F ) is characterized by the
property that it is closed with respect to extensions, direct sums, and factor modules.
If, in addition, T is closed with respect to submodules, then T and ( T, F ) are said
to be hereditary. If T is also a torsion-free class, i.e., closed with respect to products,
there is another torsion theory ( C, T ). Then T is called a TTF class [10]. Every
torsion class T gives rise to a radical, that is, an endofunctor T of R-Mod with
T M T M/ ( )( ) = 0 for all M ∈ R-Mod. Namely, T M( ) is the largest submodule T ≤
≤ M with T ∈ T. In this way, the (hereditary) torsion theories correspond to the
idempotent (left exact) radicals [11].
For any full subcategory C of R-Mod, the torsion class T C( ) generated by C
is defined to be the smallest torsion class T with C T⊂ . If C is closed with respect
to factor modules, then T C( ) consists of the modules M such that each non-zero
factor module of M has a non-zero submodule in C . If C is also closed with respect
to submodules, then T C( ) is hereditary ([11|, VI, Propositions 2.5 ana 3.3). Note that
a hereditary torsion class T is determined by its Gabriel filter, i.e., the set F of left
ideals I with R I/ .∈T
In what follows, let R-ss denote the full subcategory of semisimple modules in R-
© W. RUMP, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7 1005
1006 W. RUMP
Mod. An R-module is said to be semi-artinian if it belongs to the torsion class
generated by R-ss. By the above remark, this torsion class is hereditary. The ring R
is called left semi-artinian if R R is so. A hereditary torsion class T consisting of
semi-artinian modules is generated by T ∩ R -ss, hence by just one semisimple
module. We call such a torsion class semisimple (see also [2], with slight
modification of the terminology in [11]). Recall that R is said to be right perfect [12]
if every right R-module has a projective cover. The following proposition is
essentially well-known (cf. [11], VIII, Corollary 6.3).
Proposition 1. A ring R is right perfect if and only if every hereditary torsion
class in R-Mod is a semisimple TTF class.
For any left ideal I of R, consider the torsion class
T l M R IM M:= ∈ ={ }-Mod . (1)
In accordance with O. Horbachuk (see [5]), who studied the radical of T I , we call T I
an I-torsion class. A torsion theory ( T, F ) with T = T I will be called an I-
torsion theory. Since IM = IRM, we may assume, without loss of generality, that I
is an ideal. Thus if T I is hereditary, the corresponding Gabriel filter T I is given by
T I = H R I H RR≤ + ={ }. (2)
More generally, a hereditary torsion class with a Gabriel filter (2) for some left ideal I
is called an S-torsion class [2, 7, 13]. Thus we obtain the following proposition.
Proposition 2. An I-torsion class is hereditary if and only if it is an S - torsion
class qiven by a (two-sided) ideal.
Not every left ideal I of R defines an S-torsion class. However, any left ideal I
defines a multiplicative submonoid 1 + I of R. Hence
T
I M R x M a I a x: : ( )= ∈ ∀ ∈ ∃ ∈ + ={ }-Mod 1 0 (3)
is a hereditary torsion class in R-Mod. In fact, have the following proposition.
Proposition 3. Every S-torsion class in R-Mod is of the form (3). Precisely,
a left ideal I of R defines an S-torsion class if and only if 1 + I satisfies the left
Ore condition.
Proof. Let I be a left ideal of R. Then a left ideal H satisfies R H I/ ∈T if
and only if for each a R∈ , there is an element b I∈ with ( )1 + ∈b a H . The latter
condition means that 1 + b ∈ ( : )H a . Therefore, H belongs to the Gabriel filter of
T I if and only if I + ( : )H a = R for all a R∈ . Thus if I defines an S-torsion class
T, then T = T I . Moreover, we infer that R H I/ ∈T implies I + H = R. Hence I
defines an S-torsion class if and only if the reverse implication
I H R R H I+ = ⇒ ∈/ T (4)
holds for all left ideals H. Explicitly, condition (4) states that if 1 + a ∈ H holds for
some a I∈ , then each x R∈ satisfies ( 1 + b ) x ∈ )H for some b I∈ . In other
words, for any a I∈ and x R∈ , there exists an element b I∈ with ( 1 + b ) x ∈
∈)R ( l + a ). But this is just the left Ore condition for 1 + I.
The proposition is proved.
Let us define a torsion sequence to be a sequence T T0, ,…( )n of full sub-
categories T i R⊂ -Mod such that T Ti i−( )1, a torsion theory for i ∈){1, … , n}. For
n = 2, this means that T1 is a TTF-class, and then T T T0 1 2, ,( ) is said to be a TTF-
theory.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
I-RADICALS AND RIGHT PERFECT RINGS 1007
For any torsion class T with radical T, the ideal I : = T R( ) of R satisfies
IM T M⊂ ( ) (5)
for all M ∈ R-Mod. The following proposition shows that TTF-theories correspond to
a particular class of I-radicals.
Proposition 4. Let T be a torsion class in R-Mod with radical T and I : =
: = T R( ) . The following are equivalent :
(a) there is a TTF-theory T F D, ,( ) in R-Mod;
(b) T is an I-torsion class;
(c) T M IM( ) = for all M R∈ -Mod.
Proof. (a) ⇒ (b): For a given M ∈T , consider an epimorphism p : F →→ M
with a free R-module F. Then p IF( ) = IM. Hence M IM/ is an epimorphic image
of F IF/ ∈F . Since F is a torsion class, M IM/ ∈F T∩ , and thus M IM/ = 0.
Conversely, M = IM implies M ∈T by (5).
(b) ⇒ (c): By (5), we have T M( ) = IT M IM T M( ) ( )⊂ ⊂ .
(c) ⇒ (a): There is a torsion theory T F,( ) such that F is closed with respect to
factor modules. Hence F is a TTF class.
Corollary 1. There is a one-to-one correspondence between TTF-theories in R-
Mod and I-torsion classes given by an idempotent ideal.
Proof. An I-torsion class given by an ideal I = I2 satisfies (c) of Proposition 4.
This establishes the correspondence.
A torsion theory T T1 2,( ) in R-Mod is said to be centrally splitting if R = R1 ×
× R2 such that T i coincides with the full subcategory Ri -Mod of R-Mod.
Proposition 5. For a torsion theory T F,( ) in R-Mod, the following are
equivalent :
(a) T F,( ) is centrally splitting;
(b) F T,( ) is a torsion theory;
(c) there is a torsion sequence B C T F D, , , ,( ) .
Proof. The implications (a) ⇒ (b) ⇒ (c) are trivial. Thus let (c) be satisfied.
Consider a morphism f : F → T with F ∈F and T ∈T . Then f F( ) ∈F T∩ = 0.
Hence HomR F T,( ) = 0, which gives F C⊂ . Similarly, we get HomR T C,( ) = 0,
which yields C F⊂ . Hence C = F. Let T be the radical of T and F the radical of
F with respect to the torsion theory F T,( ) . Then I : = T R( ) and J : = F R( ) are
ideals of R. For any M R∈ -Mod, Proposition 4 implies that M JM/ = I M JM/( ) =
= IM JM JM+( ) / , whence M = IM ⊕ JM. Thus (a) holds.
Now we are ready to prove our main result. Recall that a ring R with Jacobson
radical Rad R is said to be quasilocal if R R/ Rad is a simple artinian ring.
Theorem 1. For a ring R, the following are equivalent :
(a) every hereditary torsion class in R-Mod is an I-torsion class;
(b) every hereditary torsion class in R-Mod is an S-torsion class;
(c) R = R1 × … × Rn with quasilocal right perfect rings Ri .
Proof. The implication (a) ⇒ (b) follows by Proposition 2.
(b) ⇒ (c): By [2], Corollary 3, the ring R is right perfect. Thus let P be an
indecomposable projective left R-module, and let T be the torsion class generated by
the simple R-module S : = P P/ Rad . By Proposition 3, there exists a left ideal I of
R with T = T I . Moreover, Proposition 1 implies that there is a torsion theory C T,( )
in R-Mod. So there is a smallest submodule M of P such that P M/ ∈T . Choose
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
1008 W. RUMP
e P P∈ \ Rad . Then (3) implies that (1 + a ) e ∈ Rad P for some a I∈ . Hence P =
= Re ⊂ Ie + Rad P , and thus Ie = P. Let p : R →→ P denote the epimorphism given
by the right multiplication p x( ) : = xe. Then p I( ) = Ie = P, whence I + Ker p = R.
By (2), we get P ≅ R p/ Ker ∈T . On the other hand, let ′P be an indecomposable
projective R-module with ′ ′ /≅P P S/ Rad . Then ′ ′ ∉P P/ Rad T , which yields
′ ∈P C . Consequently, we get a decomposition RR = Pm ⊕ Q with P
m ∈T and
Q ∈C . Thus Hom ( , )R
mQ P = 0. By symmetry, this implies that Hom ( , )R P P′ = 0
for each indecomposable projective R-module ′ /≅P P . Hence Hom ( , )R
mP Q = 0.
Since R RR≅ End op( ) , we get a ring decomposition R = R1 × ′R1 with R1 ≅ Pm as
left R-modules. By induction, this gives the desired decomposition R = R1 × … × Rn .
(c) ⇒ (a): Since R is right perfect, every hereditary torsion class T in R-Mod is
semisimple by Proposition 1. Hence T defines a centrally splitting torsion theory.
Now Proposition 4 completes the proof.
As a consequence, we get a relationship between left and right I-radicals (cf. [5],
Theorem 5). Recall that a torsion theory T F,( ) in R-Mod is said to be splitting if
every M R∈ -Mod admits a decomposition M = T ⊕ F with T ∈T and F ∈F .
Corollary 2. For a right perfect ring R, the following are equivalent :
(a) every hereditary torsion class in R-Mod is an I-torsion class ;
(b) every I-torsion theory in Mod-R splits.
Proof. (a) ⇒ (b): By Theorem 1, we have R = R1 × … × Rn with quasilocal
Ri . So we can assume, without loss of generality, that R is quasilocal. Then it
suffices to show that there is no nontrivial I-torsion theory in Mod-R. In fact, every
proper ideal I of R is superfluous in Mod-R by [14], Lemma 28.3. Hence I defines
the zero torsion class.
(b) ⇒ (a): Let P ∈ Mod-R be indecomposable and projective, and Rr = P Qm ⊕
with m maximal. Then I : = Pm + Rad R is an ideal of R with PI = P. As the I-
torsion theory in Mod-R splits, we infer that Pm is the I-torsion part of R, hence an
ideal. By symmetry, this yields a decomposition R = R1 × … × Rn into quasilocal
Ri . Now Theorem 1 completes the proof.
As a second consequence, we get an extension of Teply’s result (cf. [9],
Theorem 3.3).
Theorem 2. For a ring R, the following are equivalent :
(a) every hereditary torsion class in R-Mod is an I-torsion class;
(b) every left exact radical is exact;
(c) each hereditary torsion class T in R- Mod extends to a TTF-theory
T F D, ,( ) ;
(d) every hereditary torsion theory in R-Mod is centrally splitting.
Proof. (a) ⇒ (b): By Theorem 1 and Proposition 1, R decomposes into quasilocal
rings, and every hereditary torsion class in R-Mod is semisimple. Whence (b) holds.
(b) ⇒ (c): Let T F,( ) be a hereditary torsion theory in R-Mod, and let T be the
radical of T. For an epimorphism M N→→ in R-Mod with M ∈F , the exactness
of T implies that T N( ) = 0. Hence F is a torsion class.
(c) ⇒ (d) ⇒ (a): This follows by Propositions 5 and 4, respectively.
1. Horbachuk O. L., Komarnitskiy M. Ya. I-radicals and their properties // Ukr. Math. J. – 1978. – 30,
# 2. – P. 212 – 217 (in Russian).
2. Horbachuk O. L., Maturin Yu. P. On S-torsion theories in R-Mod // Mat. Studii. – 2001. – 15, # 2.
– P. 135 – 139.
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 7
3. Horbachuk O. L., Maturin Yu. P. I-radicals, their lattices and some classes of rings // Ukr. Math. J.
– 2002. – 54, # 7. – P. 1016 – 1019.
4. Horbachuk O. L., Maturin Yu. P. Rings and properties of lattices of I-radicals // Bul. Acad. Sci.
Rep. Moldova. – 2002. – 38, # 1. – P. 44 – 52.
5. Horbachuk O. L., Maturin Yu. P. On I-radicals // Ibid. – 2004. – 2(45). – P. 89 – 94.
6. Horbachuk O. L. Talk at the 5th Int. Algebraic Conf. In Ukraine. – Odessa, 2005.
7. Komarnitskiy M. Ya. Duo rings over which all torsions are S-torsions // Mat. Issled. – 1978. – 48. –
P. 65 – 68 (in Russian).
8. Dickson S. E. A torsion theory for abelian categories // Trans. Amer. Math. Soc. – 1966. – 121. –
P. 223 – 235.
9. Teply M. L. Homological dimension and splitting torsion theories // Pacif. J. Math. – 1970. – 34. –
P. 233 – 205.
10. Jans J. P. Some aspects of torsion // Ibid. – 1965. – 15. – P. 1249 – 1259.
11. Stenström B. Rings of quotients. – New York etc.: Springer, 1975.
12. Bass H. Finitistic dimension and a homological generalization of semi-primary rings // Trans.
Amer. Math. Soc. – 1960. – 95. – P. 466 – 488.
13. Golan J. S. On some torsion theories studied by Komarnickiy // Houston J. Math. – l981. – 7 . –
P. 239 – 247.
14. Anderson F. W., Fuller K. R. Rings and categories of modules. – New York etc.: Springer, 1974.
Received 09.08.2005
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| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| spelling | umjimathkievua-article-33632020-03-18T19:52:14Z I-radicals and right perfect rings I-радикали та досконалі справа кільця Rump, W. Рамп, У. We determine the rings for which every hereditary torsion theory is an S-torsion theory in the sense of Komarnitskiy. We show that such rings admit a primary decomposition. Komarnitskiy obtained this result in the special case of left duo rings. Визначено кільця, для яких кожна теорія скруту з успадкуванням є теорією S-скруту у сенсі Комарницького. Показано, що такі кільця допускають первинний розклад. Комарницький отримав цей результат у частинному випадку лівих дуо-кілець. Institute of Mathematics, NAS of Ukraine 2007-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3363 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 7 (2007); 1005–1008 Український математичний журнал; Том 59 № 7 (2007); 1005–1008 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3363/3469 https://umj.imath.kiev.ua/index.php/umj/article/view/3363/3470 Copyright (c) 2007 Rump W. |
| spellingShingle | Rump, W. Рамп, У. I-radicals and right perfect rings |
| title | I-radicals and right perfect rings |
| title_alt | I-радикали та досконалі справа кільця |
| title_full | I-radicals and right perfect rings |
| title_fullStr | I-radicals and right perfect rings |
| title_full_unstemmed | I-radicals and right perfect rings |
| title_short | I-radicals and right perfect rings |
| title_sort | i-radicals and right perfect rings |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3363 |
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