Some results on MP-injectivity and MGP-injectivity of rings and modules
We study MP-injective rings and MGP-injective rings satisfying some additional conditions. Using the concepts of MP-injectivity and MGP-injectivity of rings and modules, we present some new characterizations of QF-rings, semisimple Artinian rings, strongly regular rings, and simple Artinian rings.
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| author | Zhanmin, Zhu Чжанмін, Чжу |
| author_facet | Zhanmin, Zhu Чжанмін, Чжу |
| author_sort | Zhanmin, Zhu |
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| datestamp_date | 2020-03-18T19:37:09Z |
| description | We study MP-injective rings and MGP-injective rings satisfying some additional conditions.
Using the concepts of MP-injectivity and MGP-injectivity of rings and modules, we present some new characterizations of QF-rings,
semisimple Artinian rings, strongly regular rings, and simple Artinian rings. |
| first_indexed | 2026-03-24T02:30:52Z |
| format | Article |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 512.5
Zhanmin Zhu (Jiaxing Univ., China)
SOME RESULTS ON MP-INJECTIVITY
AND MGP-INJECTIVITY OF RINGS AND MODULES
ДЕЯКI РЕЗУЛЬТАТИ ПРО MP-IН’ЄКТИВНIСТЬ
ТА MGP-IН’ЄКТИВНIСТЬ КIЛЕЦЬ ТА МОДУЛIВ
We study MP-injective rings and MGP-injective rings satisfying some additional conditions. Using the concepts
of MP-injectivity and MGP-injectivity of rings and modules, we present some new characterizations of QF-
rings, semisimple Artinian rings, strongly regular rings, and simple Artinian rings.
Вивчаються MP-iн’єктивнi та MGP-iн’єктивнi кiльця, що задовольняють деякi додатковi умови. Iз засто-
суванням понять MP-iн’єктивностi та MGP-iн’єктивностi кiлець та модулiв наведено новi характеризацiї
QF-кiлець, напiвпростих кiлець Артiна, сильно регулярних кiлець та простих кiлець Артiна.
1. Introduction. Throughout this article, R is an associative ring with an identity. For
a subset X of R, the right and left annihilators of X are denoted by r(X) and l(X),
respectively. To facilitate, r(a) is called a special right annihilator of R for each a ∈ R.
The Jacobson radical of R is denoted by J = J(R), the right singular ideal of R is
denoted by Zr = Z(RR). The right socle of R is denoted by Sr = Soc(RR). Let M
be an R-module and N be a submodule of M, following [1], we write N ⊆ess M to
indicate thatN is an essential submodule ofM. Concepts which have not been explained
can be found in [1] and [2].
Recall that a ring R is right P-injective [3] if every R-homomorphism from a princi-
pal right ideal of R to R extends to an endomorphism of R. A ring R is right generalized
principally injective (briefly right GP-injective) [4] if, for any 0 6= a ∈ R, there exists
a positive integer n such that an 6= 0 and any right R-homomorphism from anR to R
extends to an endomorphism of R. GP-injective rings are studied in papers [4 – 8]. In
[8], GP-injective rings are called YJ-injective rings.
In [2], the concepts of right P-injective rings and right GP-injective rings are gen-
eralized to right MP-injective rings and right MGP-injective rings, respectively, and
some interesting results on these rings are obtained. Following [2], a right R-module
N is MP-injective if, for every R-monomorphism from a principal right ideal of R
to N extends to a homomorphism of R to N, the ring R is right MP-injective if RR
is MP-injective; a right R-module N is MGP-injective if, for any 0 6= a ∈ R, there
exists a positive integer n such that an 6= 0 and any R-monomorphism from anR to
N extends to a homomorphism of R to N, the ring R is right MGP-injective if RR
is MGP-injective. In this paper, we shall study some new properties of MP-injective
rings and MGP-injective rings, and give some new characterizations of QF-rings rings,
semisimple artinian rings, von Neumann regular rings, strongly regular rings and simple
artinian rings by MP-injectivity and MGP-injectivity of rings and modules.
2. Results. Recall that a ring R is QF if it is right or left self-injective and right or
left artinian, a ring R is semiregular if R/J(R) is von Neumann regular and idempotents
can be lifted modulo J(R), a ring R is right CF if every cyclic right R-module embeds in
c© ZHANMIN ZHU, 2011
1426 ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 10
SOME RESULTS ON MP-INJECTIVITY AND MGP-INJECTIVITY . . . 1427
a free module, a ring R is right mininjective if every R-homomorphism from a minimal
right ideal of R to R extends to an endomorphism of R. These concepts can be found
in [1]. It is well known that right CF-rings are left P-injective [1] (Lemma 7.2 (1));
and a ring R is QF if and only if R is right artinian and right and left mininjective
[9] (Corollary 4.8). According to [10], a ring R is right 2-simple injective if every R-
homomorphism from a 2-generated right ideal of R to R with simple image extends to
an endomorphism of R.
Theorem 2.1. Let R be a right MGP-injective ring. Then the following statements
are equivalent:
(1) R is a QF-ring;
(2) R is a right 2-simple injective ring with ACC on right annihilators;
(3) R is right CF-ring and the ascending chain r(a1) ⊆ r(a2a1) ⊆ r(a3a2a1) ⊆ . . .
terminates for every sequence {a1, a2, . . .} ⊆ R;
(4) R is a semiregular right CF-ring.
Proof. (1)⇒ (2). Since a QF-ring is right self-injective and right noetherian, so (1)
implies (2).
(2)⇒ (1). Suppose (2) holds. Then since R is a right MGP-injective ring with ACC
on right annihilators, by [2] (Corollary 3.12(1)), R is semiprimary. Noting that R is right
2-simple injective, by [10] (Theorem 17(17)), R is a QF-ring.
(1)⇒ (3). Assume (1). Then since every injective module over a QF-ring is projec-
tive, so every right R-module embeds in a free module, and hence R is a right CF-ring.
Note that a QF-ring is right noetherian, the last assertion of (3) is clear.
(3)⇒ (4). By [2] (Theorem 3.11), R is right perfect, so that it is semiregular.
(4) ⇒ (1). Note that right MGP-injectivity implies that J(R) = Zr by [2] (The-
orem 3.4(2)), so R is right artinian by [11] (Corollary 2.9). Since R is right and left
mininjective, by [9] (Corollary 4.8), R is QF.
Theorem 2.1 is proved.
Corollary 2.1 ([12], Corollary 3). The following statements are equivalent for a
ring R:
(1) R is a QF-ring;
(2) R is a right 2-injective ring with ACC on right annihilators.
Lemma 2.1. Let R be a left noetherian ring. If I is an ideal of R and r(I) ⊆ess RR,
then I is nilpotent.
Proof. Since R is left noetherian and r(Ii) is an ideal for each positive integer i,
there exists k ≥ 1 such that r(Ik) = r(Ik+1) = . . . . If I is not nilpotent, choose l(x)
maximal in {l(y) | Iky 6= 0}. Then I2kx 6= 0 because r(I2k) = r(Ik), so there exists
a ∈ Ik such that Ikax 6= 0. Since r(I) ⊆ r(Ik) and r(I) ⊆ess RR, we have that
r(Ik) ⊆ess RR. Thus axR ∩ r(Ik) 6= 0, say 0 6= axb ∈ r(Ik), then, Ikxb 6= 0 and
Ika ⊆ l(xb) but Ika * l(x), which contradicts the maximality of l(x). Therefore I is
nilpotent.
Lemma 2.1 is proved.
Theorem 2.2. Let R be a left noetherian right MGP-injective ring. Then:
(1) r(J) ⊆ess RR;
(2) J is nilpotent;
(3) r(J) ⊆ess
RR;
(4) lr(J) = J.
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1428 ZHANMIN ZHU
Proof. (1). Let 0 6= x ∈ R. Since R is left noetherian, the non-empty set F =
= {l((xa)k) | a ∈ R, k > 0 such that (xa)k 6= 0} has a maximal element, say l((xy)n).
We claim that J(xy)n = 0. If not, then there exists t ∈ J such that t(xy)n 6= 0. Since
R is right MGP-injective, there exists a positive integer m such that (t(xy)n)m 6= 0 and
b ∈ R(t(xy)n)m for every b ∈ R with r((t(xy)n)m) = r(b). Write (t(xy)n)m =
= s(xy)n, where s = (t(xy)n)m−1t ∈ J. We proceed with the following two cases.
Case 1: r((xy)n) = r(s(xy)n). Then (xy)n = cs(xy)n, i.e., (1 − cs)(xy)n = 0.
Since s ∈ J, 1− cs is invertible. So we have (xy)n = 0. This is a contradiction.
Case 2: r((xy)n) 6= r(s(xy)n). Then there exists u ∈ r(s(xy)n) but u /∈ r((xy)n).
Thus, s(xy)nu = 0 and (xy)nu 6= 0. This shows that s ∈ l((xy)nu) and l((xy)nu) ∈ F.
Noting that s /∈ l((xy)n), so the inclusion l((xy)n) ⊂ l((xy)nu) is strict. This contracts
the maximality of l((xy)n) in F.
Thus, J(xy)n = 0, and so 0 6= (xy)n ∈ xR ∩ r(J), proving (1).
(2). By (1) and Lemma 2.1.
(3). If 0 6= c ∈ R, we must show that Rc ∩ r(J) 6= 0. This is clear if Jc = 0.
Otherwise, since J is nilpotent by (2), there exists m ≥ 1 such that Jmc 6= 0 but
Jm+1c = 0. Then 0 6= Jmc ⊆ Rc ∩ r(J), as required.
(4). By (1) and [2] (Theorem 3.4), lr(J) ⊆ Zr = J, so that lr(J) = J.
Theorem 2.2 is proved.
Theorem 2.3. Let R be a left noetherian right MGP-injective ring. Then the fol-
lowing statements are equivalent:
(1) R is right Kasch;
(2) R is left C2;
(3) R is left GC2;
(4) R is semilocal;
(5) R is left artinian;
(6) the ascending chain r(a1) ⊆ r(a2a1) ⊆ r(a3a2a1) ⊆ . . . terminates for every
sequence {a1, a2, . . .} ⊆ R.
Proof. (1)⇒ (2). By [1] (Proposition 1.46).
(2)⇒ (3); and (5)⇒ (6) are obvious.
(3) ⇒ (4). Since left noetherian ring is left finite dimensional, and left finite dimen-
sional left GC2 ring is semilocal [13] (Lemma 1.1), so (4) follows from (3).
(4) ⇒ (5). Since R is left noetherian right MGP-injective, by Theorem 2.2(2), J is
nilpotent. And so R is left noetherian and semiprimary by hypothesis, as required.
(5) ⇒ (1). Assume (5). Then R is semiperfect right mininjective ring and
Sr ⊆ess RR. So that R is a right minfull ring. By [1] (Theorem 3.12), R is right
Kasch.
(6)⇒ (4). By [2] (Theorem 3.11).
Theorem 2.3 is proved.
Corollary 2.2. Let R be a left noetherian right MGP-injective right finite dimen-
sional ring. Then R is left artinian.
Proof. Since R is right MGP-injective, by [2] (Theorem 3.4(1)), R is right GC2. But
right finite dimensional right GC2 ring is semilocal, so R is left artinian by Theorem 2.3.
Corollary 2.3. The following statements are equivalent for a ring R:
(1) R is a QF-ring;
(2) R is left artinian and right 2-injective;
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SOME RESULTS ON MP-INJECTIVITY AND MGP-INJECTIVITY . . . 1429
(3) R is left noetherian right 2-injective and right Kasch;
(4) R is a left noetherian right 2-injective semilocal ring;
(5) R is left noetherian right 2-injective and left C2;
(6) R is left noetherian right 2-injective and left GC2;
(7) R is left noetherian right 2-injective and the ascending chain r(a1) ⊆ r(a2a1) ⊆
⊆ r(a3a2a1) ⊆ . . . terminates for every sequence {a1, a2, . . .} ⊆ R;
(8) R is left noetherian right 2-injective and right finite dimensional.
Proof. By Theorem 2.3 (2) through (7) are equivalent. (1)⇒ (8) is clear. (8)⇒ (2)
by Corollary 2.2. (2) ⇒ (1) by [10] (Theorem 17).
Lemma 2.2. Let M be a right R-module and NR ⊆ess MR. Then (N : x) ⊆ess RR
for all x ∈M, where (N : x) = {a ∈ R | xa ∈ N}.
Proof. Let x ∈ M. For each 0 6= a ∈ R, if xa = 0, then a ∈ (N : x), thus
0 6= aR = (N : x) ∩ aR. If xa 6= 0, then since N ⊆ess M, N ∩ xaR 6= 0, so that there
exists 0 6= xar ∈ N, and thus 0 6= ar ∈ (N : x) ∩ aR. Hence, (N : x) ⊆ess RR.
Lemma 2.2 is proved.
Theorem 2.4. The following conditions are equivalent for a ring R :
(1) R is a semisimple artinian ring;
(2) R is right Kasch and every simple right R-module is MGP-injective;
(3) R is right Kasch and every simple right R-module is mininjective.
Proof. It is obvious that (1)⇒ (2)⇒ (3).
(3) ⇒ (1). For any right R-module A, let E(A) be the injective hull of A. If A 6=
6= E(A), then there exists x ∈ E(A) − A. By Lemma 2.2, we have (A : x) ⊆ess RR.
Clearly, (A : x) 6= R. Thus there exists a maximal right ideal M of R such that
(A : x) ⊆ M. Clearly, M ⊆ess RR. Since R is right Kasch, there exists 0 6= a ∈ R
such that M = r(a). Now we define f : aR → R/r(a); ay 7→ y + r(a), then f is a
right R-homomorphism. Since aR is a minimal right ideal and R/r(a) is a simple right
R-module, by hypothesis, there is b ∈ R such that 1 + r(a) = f(a) = ba+ r(a), which
yields that 1 − ba ∈ r(a), and so a = aba. Let e = ba, then 0 6= e = e2. It follows
that M = r(e) = (1 − e)R, and then M ∩ eR = 0 but eR 6= 0, which contradicts
that M ⊆ess RR. Hence, A = E(A), i.e., A is injective. Therefore R is a semisimple
artinian ring.
Theorem 2.4 is proved.
The following Lemma 2.3 (1) and (2) are well-known results, we give their proof
here for completeness.
Lemma 2.3. Let R be a prime ring, then:
(1) if I is a nonzero ideal of R, then I is essential in R both as a left ideal and as a
right ideal;
(2) if R is a semisimple artinian ring, then it is a simple artinian ring;
(3) if R satisfies the ascending chain condition for special right annihilators, then
Zr = 0.
Proof. (1). If K is a right ideal of R satisfies K ∩ I = 0, then KI ⊆ K ∩ I = 0.
Since R is a prime ring, K = 0, and so I is an essential right ideal of R. Similarly, I is
an essential left ideal of R.
(2). Let I be a nonzero ideal of R. Then since R is a semisimple artinian ring, there
exists a right ideal T of R such that I ⊕ T = R. By (1), T = 0, and thus I = R. This
proves that R is a simple artinian ring.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 10
1430 ZHANMIN ZHU
(3). Since R satisfies the ascending chain conditions for special right annihilators,
the set {r(x) | 0 6= x ∈ R} has a maximal element r(a). If Zr 6= 0, then aZra 6= 0
because R is a prime ring (otherwise, if aZra = 0, then aZr(aR) = 0, and so aZr = 0,
i.e., (Ra)Zr = 0, which implies that Zr = 0, a contradiction). Thus there is b ∈ Zr
such that aba 6= 0. It follows from the maximality of r(a) that r(a) = r(aba). Since
aba ∈ Zr, we have r(a) = r(aba) ⊆ess RR, and whence r(a) ∩ baR 6= 0. So that there
exists c ∈ R such that bac 6= 0 and abac = 0, which implies that c ∈ r(aba) = r(a).
Thus ac = 0, and then bac = 0 which contradicts bac 6= 0. Therefore Zr = 0.
Lemma 2.3 is proved.
Theorem 2.5. The following statements are equivalent for a ring R:
(1) R is a simple artinian ring;
(2) R is a right MGP-injective prime ring such that the ascending chain r(a1) ⊆
⊆ r(a2a1) ⊆ r(a3a2a1) ⊆ . . . terminates for every sequence {a1, a2, . . .} ⊆ R;
(3) R is a prime ring such that 0 6= Sr is MP-injective, and R satisfies the ascending
chain condition for special right annihilators.
Proof. It is obvious that (1)⇒ (2) and (3).
(2)⇒ (1). By [2] (Theorem 3.17) and Lemma 2.3(2).
(3) ⇒ (1). We first prove that R is a semisimple artinian ring. If not, then Sr 6= R.
Since R satisfies the ascending chain conditions for special right annihilators, the set
{r(x) | x ∈ R − Sr} has a maximal element r(a). By Lemma 2.3(3), there exists a
nonzero right ideal T of R such that r(a)⊕T ⊆ess RR. By Lemma 2.3(1), T ∩Sr 6= 0,
so that there exists 0 6= b ∈ T ∩ Sr. Now we define f : abR → Sr; abx 7→ bx, then f
is a right R-monomorphism. Since Sr is MP-injective, then there is y ∈ Sr such that
b = f(ab) = yab, which implies that (a−aya)b = 0, i.e., b ∈ r(a−aya). Since a /∈ Sr
and y ∈ Sr, a− aya /∈ Sr. By the maximality of r(a), we have r(a) = r(a− aya). It
follows that ab = 0, and so b = yab = 0, which contradicts b 6= 0. Therefore, Sr = R,
i.e., R is a semisimple artinian ring. Since R is a prime ring, by Lemma 2.3(2), R is a
simple artinian ring.
Theorem 2.5 is proved.
Recall that a ring R is a right SF ring if every simple right R-module is flat, a ring
R is a right quasi-duo ring if every maximal right ideal of R is an ideal, a ring R is a
quasi-duo ring if it is left or right quasi-duo. These concepts can be found in [14].
Proposition 2.1. If every maximal essential right ideal of R is MP-injective, then
R is a right SF ring.
Proof. Let S be a simple right R-module, then there exists a maximal right ideal
M of R such that S ∼= R/M. If M is essential right ideal, then by hypothesis, M
is MP -injective. So for any a ∈ R, if y = xa ∈ Ra ∩M, then since the inclusion
mapping yR → M extends to a right R-homomorphism f : R → M, so that y =
= f(y) = f(1)y = (f(1)x)a ∈ Ma. Hence Ra ∩M = Ma, this shows that M is a
pure submodule of R, and therefore R/M is flat.
Proposition 2.1 is proved.
Definition 2.1. Let R be a ring. A right R-module N is WMGP-injective if, for
any a ∈ R, there exists a positive integer n such that any R-monomorphism from anR
to N extends to a homomorphism of R to N. The ring R is right WMGP-injective if RR
is WMGP-injective.
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SOME RESULTS ON MP-INJECTIVITY AND MGP-INJECTIVITY . . . 1431
Example 2.1. Let R =
{[
a v
0 a
]∣∣∣∣ a ∈ F, v ∈ V} be the trivial extension of the
field F by the two-dimensional vector space V over F. Then R is a commutative WMGP-
injective ring that is not MGP-injective.
Proof. Let V = uF ⊕ wF. For any x ∈ R, write x =
[
a v
0 a
]
. If a 6= 0, then x
is invertible, so xR = R, and thus any R-homomorphism from xR to R extends to an
endomorphism of R. If a = 0, then x2 = 0, and so any R-homomorphism from x2R to
R extends to an endomorphism of R. Hence, R is WMGP-injective. Let x0 =
[
0 u
0 0
]
,
y0 =
[
0 w
0 0
]
, then x02 = 0, r(x0) = r(y0) =
[
0 V
0 0
]
, but Rx0 =
[
0 Fu
0 0
]
and
Ry0 =
[
0 Fw
0 0
]
. So Ry0 * Rx0. This shows that the R-monomorphism from x0R
to R via x0r 7→ y0r can not be extended to an endomorphism of R. whence R is not
MGP-injective.
Proposition 2.1 is proved.
Next, we give some new characterizations of strongly regular rings.
Theorem 2.6. The following conditions are equivalent for a ring R :
(1) R is a strongly regular ring;
(2) every maximal right ideal of R is MGP-injective and l(a) is an ideal for each
a ∈ R;
(3) R is a reduced ring and every maximal essential right ideal of R is MGP-
injective;
(4) R is a reduced ring and every maximal essential right ideal of R is WMGP-
injective or a right annihilator;
(5) R is a quasi-duo ring, and every maximal essential right ideal of R is MP-
injective.
Proof. (1)⇒ (2). Since R is a strongly regular ring, by [15] (Proposition 12.3), R
is von Neumann regular and every left ideal is two-sided, so (2) holds.
(2) ⇒ (3). We need only to prove that R is reduced. Let a ∈ R with a2 = 0, we
claim that a = 0. Otherwise, if a 6= 0, then a ∈ l(a) 6= R. By (2), l(a) is an ideal, so
there exists a maximal right ideal M such that l(a) ⊆ M. Since M is MGP-injective,
the inclusion mapping aR → M extends to a homomorphism from R to M, and so
there exists b ∈ M such that a = ba. Thus 1 − b ∈ l(a) ⊆ M, and then 1 ∈ M, a
contradiction. Therefore a = 0, and hence R is reduced.
(3) ⇒ (4). Since right MGP-injective module is right WMGP-injective, so (3)
implies (4).
(4)⇒ (1). For any a ∈ R, we claim that aR+r(a) = R. In fact, if aR+r(a) 6= R,
then there exists a maximal right ideal M of R such that aR + r(a) ⊆ M. We claim
that M is an essential right ideal. Otherwise, there there exists 0 6= b ∈ R such that
bR ∩M = 0. Since M is a maximal right ideal, bR ⊕M = R, and so M = eR for
some e2 = e ∈ R. Clearly, a = ea, i.e., 1− e ∈ l(a). Since aR is reduced, l(a) ⊆ r(a),
so that 1−e ∈ r(a) ⊆M, and hence 1 ∈M, a contradiction. Therefore M is a maximal
essential right ideal. By hypothesis, M is WMGP-injective or a right annihilator.
Case 1: If M is WMGP-injective. Then there exists a positive integer n such that
any R-monomorphism from anR to M extends to a homomorphism of R to M. Now
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1432 ZHANMIN ZHU
we define f : anR → M by f(anx) = ax, where x ∈ R, noting that R is reduced, by
[2] (Lemma 3.20), f is well-defined, and f is a right R-homomorphism, and so there
exist u ∈ M such that a = uan. Thus, 1 − uan−1 ∈ l(a) ⊆ r(a) ⊆ M, it follows that
1 ∈M, a contradiction.
Case 2: If M is a right annihilator. Then there exists 0 6= c ∈ R such that M =
= r(c). Thus, c ∈ lr(c) = l(M) ⊆ l(a) ⊆ r(a) ⊆M = r(c), so that c2 = 0. Since R is
reduced, c = 0, a contradiction too.
Therefore, these contradictions show that aR + r(a) = R. Write 1 = as+ t, where
s ∈ R, t ∈ r(a), then a = a2s+ at = a2s. Consequently, R is strongly regular.
(5)⇒ (1). By Proposition 2.1 and [14] (Theorem 4.10).
Theorem 2.6 is proved.
Theorem 2.7. If R is a right MGP-injective ring, then it is a classical quotient
ring, and so every right (left) R-module is divisible.
Proof. Let l(a) = r(a) = 0. Then l(ak) = r(ak) = 0 for every positive integer k.
By the right MGP -injectivity of R, there exists a positive integer n such that b ∈ Ran
for every b ∈ R with r(an) = r(b), in particular, 1 = can for some c ∈ R. Thus,
ancan = an, noticing that l(an) = r(an) = 0, we have anc = can = 1. Hence R is a
classical quotient ring, and so every right (left) R-module is divisible.
Theorem 2.7 is proved.
Proposition 2.2. If every maximal essential right ideal of R is WMGP-injective or
a right annihilator, then R is a classical quotient ring.
Proof. Let a be a nonzero divisor of R, i.e., l(a) = r(a) = 0. Then there exists a
right ideal K such that aR⊕K ⊆ess RR. We claim that aR⊕K = R. If not, then there
exists a maximal right ideal M such that aR ⊕K ⊆ M, and so M is WMGP-injective
or a right annihilator. If M is WMGP-injective, then there exists a positive integer n
such that every monomorphism from anR to M extends to a homomorphism of R to
M. Now define f : anR → M by f(anx) = ax, where x ∈ R, then f is well defined
as a is a nonzero divisor, and so a = f(an) = ban for some b ∈ M. This follows that
1− ban−1 ∈ l(a) = 0, and then 1 ∈M, a contradiction. If M is a right annihilator, then
since M is a maximal right ideal, there exists 0 6= t ∈ R such that M = r(t). Hence,
t ∈ lr(t) = l(M) ⊆ l(a) = 0, i.e., t = 0, a contradiction too. Thus, aR ⊕ K = R.
Write aR = eR, where e2 = e, then a = ea and e = ac for some c ∈ R, and so
a = aca. Noting that a is a nonzero divisor, we have ac = ca = 1. This shows that R
is a classical quotient ring.
Proposition 2.2 is proved.
At the end of this paper, we give an important property of semiprime right MGP-
injective rings.
Proposition 2.3. If R is a semiprime right MGP-injective ring, then R contains a
unique largest reduced ideal I, and I = rl(I) = lr(I), Z(RI) = Z(IR) = 0.
Proof. Let I =
∑
α∈A
Iα be the sum of all reduced ideals Iα of R. It may be
assumed that I 6= 0. We prove that rl(I) is reduced. Otherwise, then there exists 0 6=
6= x ∈ rl(I) such that x2 = 0.
Case 1: xR ∩ Iα = 0 for all α ∈ A. Then xRIα ⊆ xR ∩ Iα = 0 for all α ∈ A,
and so xRI = 0, xR ⊆ l(I). It follows that xRx = 0. But R is semiprime, x = 0, a
contradiction.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 10
SOME RESULTS ON MP-INJECTIVITY AND MGP-INJECTIVITY . . . 1433
Case 2: There is i ∈ A such that xR ∩ Ii 6= 0. Take 0 6= a ∈ xR ∩ Ii, then
aR is reduced. For any y ∈ r(a2), since (aya)2 = ay(a2y)a = 0 and aya ∈ aR, we
have aya = 0, and then (ay)2 = (aya)y = 0, which implies that ay = 0. Hence,
r(a2) = r(a). By the proof of [2] (Lemma 3.20), we have that r(ak) = r(a) for every
positive integer k. If a2 = 0, then a = 0, a contradiction. If a2 6= 0. Since R is right
MGP-injective, by [2] (Theorem 3.2), there exists a positive integer n such that a2n 6= 0
and a = ba2n for some b ∈ R. Write c = ba2n−2, then a = ca2. It is easy to see
that (a − aca)2 = 0, a − aca ∈ aR, so a = aca. Let e = ac, then e2 = e, a = ea,
e ∈ aR ⊆ xR. Thus, there exists d ∈ R such that e = xd, (ex)2 = ex2dx = 0. But
ex ∈ aR, so ex = 0, and whence e = e2 = exd = 0, this follows that a = 0, a
contradiction too.
Therefore, rl(I) is reduced. Noting that rl(I) is an ideal and I ⊆ rl(I), we have
I = rl(I), and so I is the unique largest reduced ideal. Since R is semiprime, it is easy
to see that r(K) = l(K) for every ideal K of R. Noting that I and l(I) are ideals, we
have lr(I) = ll(I) = rl(I) = I.
It is obvious that Z(IR) = I ∩ Zr. Assume that I ∩ Zr 6= 0, then there exists
0 6= y ∈ I ∩ Zr. Since r(y) is an essential right ideal, r(y) ∩ yR 6= 0, and so there is
0 6= yz ∈ r(y). Thus, y2z = 0, (yzy)2 = yz(y2z)y = 0. But yzy ∈ I and I is reduced,
so yzy = 0, (yz)2 = (yzy)z = 0, yz ∈ I, and hence yz = 0, which contradicts yz 6= 0.
Consequently, Z(IR) = 0. Similarly, Z(RI) = 0.
Proposition 2.3 is proved.
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Received 14.03.11
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 10
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| spelling | umjimathkievua-article-28152020-03-18T19:37:09Z Some results on MP-injectivity and MGP-injectivity of rings and modules Деякi результати про MP-iн’єктивнiсть та MGP-iн’єктивнiсть кiлець та модулiв Zhanmin, Zhu Чжанмін, Чжу We study MP-injective rings and MGP-injective rings satisfying some additional conditions. Using the concepts of MP-injectivity and MGP-injectivity of rings and modules, we present some new characterizations of QF-rings, semisimple Artinian rings, strongly regular rings, and simple Artinian rings. Вивчаються MP-iн’єктивнi та MGP-iн’єктивнi кiльця, що задовольняють деякi додатковi умови. Iз застосуванням понять MP-iн’єктивностi та MGP-iн’єктивностi кiлець та модулiв наведено новi характеризацiї QF-кiлець, напiвпростих кiлець Артiна, сильно регулярних кiлець та простих кiлець Артiна. Institute of Mathematics, NAS of Ukraine 2011-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2815 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 10 (2011); 1426-1433 Український математичний журнал; Том 63 № 10 (2011); 1426-1433 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2815/2383 https://umj.imath.kiev.ua/index.php/umj/article/view/2815/2384 Copyright (c) 2011 Zhanmin Zhu |
| spellingShingle | Zhanmin, Zhu Чжанмін, Чжу Some results on MP-injectivity and MGP-injectivity of rings and modules |
| title | Some results on MP-injectivity and MGP-injectivity of rings and modules |
| title_alt | Деякi результати про MP-iн’єктивнiсть та MGP-iн’єктивнiсть кiлець та модулiв |
| title_full | Some results on MP-injectivity and MGP-injectivity of rings and modules |
| title_fullStr | Some results on MP-injectivity and MGP-injectivity of rings and modules |
| title_full_unstemmed | Some results on MP-injectivity and MGP-injectivity of rings and modules |
| title_short | Some results on MP-injectivity and MGP-injectivity of rings and modules |
| title_sort | some results on mp-injectivity and mgp-injectivity of rings and modules |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2815 |
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