Almost MGP-Injective Rings
A ring R is called right almost MGP-injective (or AMGP-injective) if, for any 0 ≠ a ∈ R, there exists an element b ∈ R such that ab = ba ≠ 0 and any right R-monomorphism from abR to R can be extended to an endomorphism of R. In the paper, several properties of these rings are establshed and some int...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508434977259520 |
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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 | 2020-03-18T19:25:49Z |
| description | A ring R is called right almost MGP-injective (or AMGP-injective) if, for any 0 ≠ a ∈ R, there exists an element b ∈ R such that ab = ba ≠ 0 and any right R-monomorphism from abR to R can be extended to an endomorphism of R. In the paper, several properties of these rings are establshed and some interesting results are obtained. By using the concept of right AMGP-injective rings, we present some new characterizations of QF-rings, semisimple Artinian rings, and simple Artinian rings. |
| first_indexed | 2026-03-24T02:25:09Z |
| format | Article |
| fulltext |
UDC 512.5
Zhu Zhanmin (Jiaxing Univ., China)
ALMOST MGP-INJECTIVE RINGS
МАЙЖЕ МGP-IН’ЄКТИВНI КIЛЬЦЯ
A ring R is called right almost MGP-injective (or AMGP-injective for short) if, for any 0 6= a ∈ R, there exists an element
b ∈ R such that ab = ba 6= 0 and any right R-monomorphism from abR to R extends to an endomorphism of R. In this
paper, several properties of these rings are given, some interesting results are obtained. Using the concept of right AMGP-
injective rings, we present some new characterizations of QF-rings, semisimple Artinian rings and simple Artinian rings.
Кiльце R називається правим майже МGP-iн’єктивним кiльцем (або правим АМGP-iн’єктивним кiльцем), якщо
для всiх 0 6= a ∈ R iснує елемент b ∈ R такий, що ab = ba 6= 0 i будь-який правий R-мономорфiзм з abR в R
продовжується до ендоморфiзму в R. В роботi наведено деякi властивостi таких кiлець та отримано деякi цiкавi
результати. З використанням поняття АМGP-iн’єктивних кiлець наведено деякi новi характеристики QF-кiлець,
напiвпростих артiнових кiлець та простих артiнових кiлець.
1. Introduction. Throughout this paper, R is an associative ring with identity, and all modules are
unitary. As usual, J = J(R), Zl (Zr) and Sl (Sr) denote respectively the Jacobson radical, the left
(right) singular ideal and the left (right) socle of R. The left (respectively, right) annihilators of a
subset X of R is denoted by l(X)
(
respectively, r(X)
)
.
Recall that a ring R is right P-injective [1] if every R-homomorphism from a principal right
ideal of R to R extends to an endomorphism of R. A ring R is right generalized principally injective
(briefly right GP-injective) [2] 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 [2 – 6]. In [6], GP-injective rings are called YJ-injective rings. It
is easy to see that right P-injective rings are right GP-injective, but right GP-injective rings need not
be right P-injective by [5] (Example 1).
In [7], the concepts of right P-injective rings and right GP-injective rings are generalized to right
MP-injective rings and right MGP-injective rings, respectively. Following [7], a ring R is called
right MP-injective if, for every R-monomorphism from a principal right ideal of R to R extends
to an endomorphism of R; a ring R is called right 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 R extends to
a endomorphism of R. In this paper, we shall generalize the concept of right MGP-injective rings
to right AMGP-injective rings, some properties of these rings will be given, conditions under which
right AMGP-injective rings are QF-rings, semisimple Artinian rings and simple Artinian rings will
be given, respectively. And right AMGP-injective left Noertherian rings will be investigated.
2. AMGP-injective rings.
Definition 2.1. A ring R is called right almost MGP-injective (or AMGP-injective for short)
if, for any 0 6= a ∈ R, there exists an element b ∈ R such that ab = ba 6= 0 and any right
R-monomorphism from abR to R extends to an endomorphism of R.
Theorem 2.1. For a ring R, the following conditions are equivalent:
(1) R is right AMGP-injective;
(2) for any 0 6= a ∈ R, there exists b ∈ R such that ab = ba 6= 0 and c ∈ Rab for every c ∈ R
with r(ab) = r(c).
c© ZHU ZHANMIN, 2013
1476 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
ALMOST MGP-INJECTIVE RINGS 1477
Proof. (1) ⇒ (2). Let 0 6= a ∈ R. Since R is right AMGP-injective, there exists an element
b ∈ R such that ab = ba 6= 0 and every monomorphism from abR to R extends to R. Suppose that
r(ab) = r(c). Then f : abR→ R; abr 7→ cr, is a monomorphism, which extends to an endomorphism
g of R. So c = f(ab) = g(ab) = g(1)ab ∈ Rab.
(2) ⇒ (1). Let 0 6= a ∈ R. By (2), there exists b ∈ R such that ab = ba 6= 0 and c ∈ Rab
for every c ∈ R with r(ab) = r(c). Let f : abR → R be monic. Then r(ab) = r
(
f(ab)
)
, and so
f(ab) = cab for some c ∈ R. It follows that f = c·, as required.
Theorem 2.1 is proved.
It is obvious that right MGP-injective rings are AMGP-injective. Our next example shows that a
right AMGP-injective rings need not be right MGP-injective.
Example 2.1. Let M = ⊕∞
i=1Zpi , where pi is the ith prime number, and let
R =
{[
n x
0 n
]∣∣∣∣∣n ∈ Z, x ∈M
}
.
Then, by [7] (Example 3.3), R is not right MGP-injective. For any 0 6= a =
[
n x
0 n
]
∈ R. If
n 6= 0, then there exists y ∈ M such that ny 6= 0. Now let b =
[
0 y
0 0
]
, then 0 6= ab = ba =
=
[
0 ny
0 0
]
∈ J(R). If n = 0, then a ∈ J(R). Thus, by the proof of [8] (Example 3.1), for any
0 6= a ∈ R, there is a, b ∈ R, such that ba = ab 6= 0 and Ir(ba) = R(ba), and so R is right
AMGP-injective by Theorem 2.1.
Recall that a ring R is called right mininjective [9] if every R-homomorphism from a minimal
right ideal of R into R extends to R.
Theorem 2.2. Let R be right AMGP-injective. Then:
(1) R is right mininjective;
(2) J(R) ⊆ Zr.
Proof. (1). It is obvious.
(2). Let a ∈ J(R), then we will show that a ∈ Zr. If not, then there exists 0 6= b ∈ R such
that r(a) ∩ bR = 0. Clearly ab 6= 0. Since R is right AMGP-injective, there exists c ∈ R such that
abc 6= 0 and u ∈ Rabc for every u ∈ R with r(abc) = r(u). Since r(abc) = r(bc), so bc = dabc
for some d ∈ R. Thus (1− da)bc = 0. Since a ∈ J(R), 1− da is invertible, and so bc = 0. Hence
abc = 0, a contradiction.
Theorem 2.2 is proved.
We note that the ring Z of integers is right mininjective but not right AMGP-injective; so right
mininjective rings need not be right AMGP-injective.
Corollary 2.1. Let R be a right AMGP-injective ring. Suppose that, for any sequence {a1,
a2, . . .} ⊆ R, the chain r(a1) ⊆ r(a2a1) ⊆ . . . terminates. Then J(R) = Zr.
Proof. Since R is right AMGP-injective, by Theorem 2.2, J(R) ⊆ Zr. Since the chain r(a1) ⊆
⊆ r(a2a1) ⊆ . . . terminates for any sequence {a1, a2, . . .} ⊆ R, by [7] (Lemma 3.10), Zr is right
T -nilpotent, and so Zr is nil. It follows that Zr ⊆ J(R), and hence J(R) = Zr.
Lemma 2.1. Let R be right AMGP-injective. If a /∈ Zr, then the inclusion r(a) ⊂ r(a− aca) is
strict for some c ∈ R.
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1478 ZHU ZHANMIN
Proof. Since r(a) is not an essential right ideal, there exists a nonzero right ideal I of R such that
r(a)⊕ I is essential in RR. Take 0 6= b ∈ I, then ab 6= 0. By the right AMGP-injectivity, there is an
element c1 in R such that abc1 6= 0 and any right R-monomorphism from abc1R to R extends to an
endomorphism of R. Observing that bR∩r(a) = 0, we have a right R-monomorphism g : abc1R→ R
given by g(abc1r) = bc1r. Thus bc1 = cabc1 for some c ∈ R, and so bc1 ∈ r(1 − ca), whence
bc1 ∈ r(a− aca). Note that bc1 6∈ r(a), thus we have that the inclusion r(a) ⊂ r(a− aca) is strict.
Lemma 2.1 is proved.
Theorem 2.3. If R is right AMGP-injective, then the following statements are equivalent:
(1) R is right perfect;
(2) the ascending chain r(a1) ⊆ r(a2a1) ⊆ r(a3a2a1) ⊆ . . . terminates for every infinite
sequence a1, a2, a3, . . . of R.
Proof. By Corollary 2.1, Lemma 2.1 and [7] (Lemma 2.10), we can complete the proof in a
similar way to that of [7] (Theorem 2.11).
Recall that a ring R is called right Kasch [10] if every simple right R-module embeds in R,
equivalently if I(T ) 6= 0 for every maximal right ideal T of R. Left Kasch rings can be defined
similarly; a ring R is called right minfull [9] if it is semiperfect, right mininjective, and Soc(eR) 6= 0
for each local idempotent e ∈ R.
Corollary 2.2. If R is a right AMGP-injective ring with ACC on right annihilators, then:
(1) R is semiprimary;
(2) R is left and right Kasch.
Proof. (1) It is well known that Zr is nilpotent for any ring R with ACC on right annihilators.
By Theorem 2.3 and Theorem 2.2(2), R is semiprimary.
(2). By (1), R is semiprimary, so R is semiperfect with essential right socle. Noting that R
is right mininjective by Theorem 2.2(1), hence it is right minfull, and thus (2) follows from [10]
(Theorem 3.12(1)).
Corollary 2.3. Let R be a right AMGP-injective ring. Then R is right Noetherian if and only if
R is right Artinian.
Proof. Let R be a right Noetherian right AMGP-injective ring. Then by Corollary 2.2, R is a
right Noetherian semiprimary ring, and so R is right Artinian.
Corollary 2.4. Let R be a right AMGP-injective ring with ACC on right annihilators and Sl ⊆
⊆ Sr. Then R is left Artinian if and only if Sl is a finitely generated left ideal.
Proof. By Corollary 2.2, R is semiprimary. By Theorem 2.2 and [9] (Theorem 1.14(4)), Sr ⊆ Sl,
and so Sl = Sr by the hypothesis. Now the result follows from [11] (Lemma 6).
Recall that a ring R is called a left minannihilator ring [9], if every minimal left ideal K is a left
annihilator, equivalently, if lr(K) = K.
Corollary 2.5. Let R be a right AMGP-injective ring with ACC on right annihilators. If R is a
left minannihilator ring, then:
(1) R is left Artinian;
(2) R is right Artinian if and only if Sr is finitely generated as a right ideal of R.
Proof. (1). By Corollary 2.2, R is semiprimary. By [9] (Corollary 3.15), R is left finite dimen-
sional with Sl = Sr. Now, by [11] (Lemma 6), R is left Artinian.
(2). The assertion follows from (1) and [11] (Lemma 6).
Definition 2.2. A ring R is called right weakly P-injective (or right WP-injective for short) if,
for any 0 6= a ∈ R, there exists b ∈ R, such that ab = ba 6= 0 and any right R-homomorphism from
abR to R extends to an endomorphism of R.
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ALMOST MGP-INJECTIVE RINGS 1479
Theorem 2.4. For a ring R, the following conditions are equivalent:
(1) R is right WP-injective;
(2) for any 0 6= a ∈ R, there exists b ∈ R such that ab = ba 6= 0 and lr(ab) = Rab.
Proof. (1) ⇒ (2). For any 0 6= a ∈ R, since R is right WP-injective, there exists an element
b ∈ R, such that ab = ba 6= 0 and any R-homomorphism from abR to R extends to R. Now let
x ∈ lr(ab), we define f : abR → R by abr 7→ xr, then f is a well defined right R-homomorphism
and hence f extends to an endomorphism g of R. Take c = g(1), then x = cab ∈ Rab. This shows
that lr(ab) = Rab.
(2) ⇒ (1). For any 0 6= a ∈ R, by (2), there exists b ∈ R such that ab = ba 6= 0 and
lr(ab) = Rab. Suppose f ∈ HomR(abR,R), then f(ab) ∈ lr(ab), and so there exists c ∈ R such
that f(ab) = cab. Let g : R→ R;x 7→ cx, then g extends f.
Theorem 2.4 is proved.
Clearly, right GP-injective rings are both right WP-injective and right MGP-injective, and right
WP-injective rings are right AMGP-injective. It is easy to see that the ring in Example 2.1 is right
WP-injective by Theorem 2.4, but it is not right MGP-injective by [7] (Example 3.3). Hence a right
WP-injective rings need not be right GP-injective. By Theorem 2.4, we see that if R is a right
WP-injective ring, then it is a left minannihilator ring, so by Corollary 2.5, we have the following
corollary.
Corollary 2.6. Let R be a right WP-injective ring with ACC on right annihilators. Then:
(1) R is left Artinian;
(2) R is right Artinian if and only if Sr is finitely generated as a right ideal of R.
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 a free module; a ring R is called right (left)
min−CS if every minimal right (left) ideal of R is essential in a direct summand of RR (RR); a ring
R is called right min−PF ring if R is a semiperfect, right mininjective ring in which Sr ⊆ess RR
and lr(K) = K for every simple left ideal K ⊆ Re, where e2 = e is local. These concepts can be
found in [10]. It is well known that right CF-rings are left P-injective [10] (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 [12], 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 2.5. Let R be a right AMGP-injective ring. Then the following are equivalent:
(1) R is a QF-ring;
(2) R is a left mininjective ring with ACC on right annihilators;
(3) R is right min−CS, left minannihilator ring with ACC on right annihilators;
(4) R is a two-sided min−CS ring with ACC on right annihilators;
(5) R is a right 2-simple injective ring with ACC on right annihilators;
(6) R is right CF-ring and the ascending chain r(a1) ⊆ r(a2a1) ⊆ r(a3a2a1) ⊆ . . . terminates
for every sequence {a1, a2, . . .} ⊆ R;
(7) R is a semiregular right CF-ring.
Proof. It is obvious that (1) implies (2) through (5).
(2)⇒ (1). By Corollary 2.2(1), R is semiprimary, so it is a semilocal, left and right mininjective
ring with ACC on right annihilators in which Sr ⊆ess RR. By [10] (Theorem 3.31), R is a QF-ring.
(3) ⇒ (1). Since R is a semiprimary left minannihilator ring, it is a right min-PF ring with
Sr = Sl by [10] (Corollary 3.25). Then R is a right minannihilator ring by [10] (Lemma 4.4) because
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
1480 ZHU ZHANMIN
it is right min-CS. Hence R is left min-PF, again by [10] (Corollary 3.25). Now [10] (Theorem 3.38)
shows that R is QF.
(4) ⇒ (1). By Corollary 2.2(2), R is left and right Kasch, and hence Sr = Sl by [10]
(Lemma 4.5(2)) because R is left and right min-CS. Thus R is a left and right min-PF ring by
[10] (Corollary 4.6), so R is QF, again by [10] (Theorem 3.38).
(5) ⇒ (1). Suppose (5) holds. Then since R is a right AMGP-injective ring with ACC on right
annihilators, by Corollary 2.2(1), R is semiprimary. Noting that R is right 2-simple injective, by [12]
(Theorem 17(17)), R is a QF-ring.
(1)⇒ (6). Assume (1). Then since every injective module over a QF-ring is projective, 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 (6) is clear.
(6) ⇒ (7). By Theorem 2.3, R is right perfect, so that it is semiregular.
(7) ⇒ (1). Note that the right AMGP-injectivity implies that J(R) ⊆ Zr by Theorem 2.2(2).
Thus, R is right Artinian by [13] (Corollary 2.9). Since R is right and left mininjective, by [9]
(Corollary 4.8), R is QF.
Corollary 2.7. Let R be a right WP -injective ring. Then R is a QF -ring if and only if R is a
right min−CS ring with ACC on right annihilators.
Theorem 2.6. Let R be a left Noetherian right AMGP-injective ring. Then:
(1) r(J) ⊆ess RR;
(2) J is nilpotent;
(3) r(J) ⊆ess
RR.
Proof. Let 0 6= x ∈ R. Since R is left Noetherian, the nonempty set F = {l(xa) | a ∈ R such
that xa 6= 0} has a maximal element, say l(xy).
We claim that Jxy = 0. If not, then there exists t ∈ J such that txy 6= 0. Since R is right
AMGP-injective, there exists a z ∈ R such that ztxy 6= 0 and b ∈ R(ztxy) for every b ∈ R with
r(ztxy) = r(b). Write ztxy = sxy, where s = zt ∈ J. We proceed with the following two cases.
Case 1. r(xy) = r(sxy). Then xy = csxy, i. e., (1−cs)xy = 0. Since s ∈ J, 1−cs is invertible.
So we have xy = 0. This is a contradiction.
Case 2. r(xy) 6= r(sxy). Then there exists u ∈ r(sxy) but u /∈ r(xy). Thus, sxyu = 0 and
xyu 6= 0. This shows that s ∈ l(xyu) and l(xyu) ∈ F. Noting that s /∈ l(xy), so the inclusion
l(xy) ⊂ l(xyu) is strict. This contracts the maximality of l(xy) in F.
Thus, Jxy = 0, and so 0 6= xy ∈ xR ∩ r(J), proving (1).
(2). By (1) and [14] (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.
Theorem 2.6 is proved.
Theorem 2.7. Let R be a left Noetherian right AMGP-injective ring. Then the following state-
ments 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;
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
ALMOST MGP-INJECTIVE RINGS 1481
(6) the ascending chain r(a1) ⊆ r(a2a1) ⊆ r(a3a2a1) ⊆ . . . terminates for every sequence
{a1, a2, . . .} ⊆ R.
Proof. (1) ⇒ (2). By [10] (Proposition 1.46).
(2) ⇒ (3); and (5)⇒ (6) are obvious.
(3) ⇒ (4). Since left Noetherian ring is left finite dimensional, and left finite dimensional left
GC2 ring is semilocal [15] (Lemma 1.1), so (4) follows from (3).
(4)⇒ (5). Since R is left noetherian right MGP-injective, by Theorem 2.6(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 [10] (Theorem 3.12), R is right Kasch.
(6) ⇒ (4). By Theorem 2.3.
Theorem 2.7 is proved.
Theorem 2.8. Let R be a right AMGP -injective ring. Then following conditions are equivalent:
(1) R is a semisimple Artinian ring;
(2) R is a semiprime ring, and the ascending chain r(a1) ⊆ r(a2a1) ⊆ r(a3a2a1) ⊆ . . .
terminates for every sequence {a1, a2, . . .} ⊆ R.
Proof. (1) ⇒ (2) is trivial.
(2) ⇒ (1). By Theorem 2.3, R is right perfect, i.e., R/J(R) is semisimple Artinian and J(R)
is right T− nilpotent. If J(R) 6= 0, then, by [7] (Lemma 3.16), J(R) is not nil, a contradiction. So
J(R) = 0, and whence R is semisimple Artinian.
Theorem 2.9. Let R be a right AMGP-injective ring. Then following conditions are equivalent:
(1) R is a simple Artinian ring;
(2) R is a prime ring, and the ascending chain r(a1) ⊆ r(a2a1) ⊆ r(a3a2a1) ⊆ . . . terminates
for every sequence {a1, a2, . . .} ⊆ R.
Proof. (1) ⇒ (2) is obvious.
(2) ⇒ (1). By Theorem 2.8 and [14] (Lemma 2.3 (2)).
1. Nicholson W. K., Yousif M. F. Principally injective rings // J. Algebra. – 1995. – 174. – P. 77 – 93.
2. Nam S. B., Kim N. K., Kim J. Y. On simple GP-injective modules // Communs Algebra. – 1995. – 23. – P. 5437 – 5444.
3. Chen J. L., Ding N. Q. On general principally injective rings // Communs Algebra. – 1999. – 27. – P. 2097 – 2116.
4. Chen J. L., Ding N. Q. On regularity of rings // Algebra Colloq. – 2001. – 8. – P. 267 – 274.
5. Chen J. L., Zhou Y. Q., Zhu Z. M. GP -injective rings need not be P -injective // Communs Algebra. – 2005. – 33. –
P. 2395 – 2402.
6. Yue Chi Ming R. On regular rings and self-injective rings II // Glas. mat. – 1983. – 18. – P. 221 – 229.
7. Zhu Z. M. MP -injective rings and MGP -injective rings // Indian J. Pure and Appl. Math. – 2010. – 41. – P. 627 – 645.
8. Yousif M. F., Zhou Y. Q. Rings for which certain elements have the principal extension property // Algebra Colloq. –
2003. – 10. – P. 501 – 512.
9. Nicholson W. K., Yousif M. F. Mininjective rings // J. Algebra. – 1997. – 187. – P. 548 – 578.
10. Nicholson W. K., Yousif M. F. Quasi-Frobenius Rings. – Cambridge: Cambridge Univ. Press, 2003.
11. Camillo V., Yousif M. F. Continuous rings with ACC on annihilators // Can. Math. Bull. – 1991. – 34. – P. 462 – 464.
12. Zhu M Z., Chen J. L. 2-Simple injective rings // Int. J. Algebra. – 2010. – 4. – P. 25 – 37.
13. Chen J. L., Li W. X. On artiness of right CF rings // Communs Algebra. – 2004. – 32. – P. 4485 – 4494.
14. Zhu Z. M. Some results on MP -injectivity and MGP -injectivity of rings and modules // Ukr. Math. J. – 2012. – 63,
№ 12. – P. 1623 – 1632.
15. Zhou Y. Q. Rings in which certain right ideals are direct summands of annihilators // J. Aust. Math. Soc. – 2002. –
73. – P. 335 – 346.
Received 29.06.12
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
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| id | umjimathkievua-article-2527 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:25:09Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/3a/5c2e766f8d3b677b8e75f997faea643a.pdf |
| spelling | umjimathkievua-article-25272020-03-18T19:25:49Z Almost MGP-Injective Rings Майже MGP-iн'єктивнi кiльця Zhanmin, Zhu Чжанмін, Чжу A ring R is called right almost MGP-injective (or AMGP-injective) if, for any 0 ≠ a ∈ R, there exists an element b ∈ R such that ab = ba ≠ 0 and any right R-monomorphism from abR to R can be extended to an endomorphism of R. In the paper, several properties of these rings are establshed and some interesting results are obtained. By using the concept of right AMGP-injective rings, we present some new characterizations of QF-rings, semisimple Artinian rings, and simple Artinian rings. Кільце R називається правим майже MGP-ін'єктивним кільцем (або правим AMGP-ін'єктивним кільцем), якщо для всіх 0 ≠ a ∈ R існує елемент b ∈ R такий, що ab = ba ≠ 0 i будь-який правий R-мономорфізм з abR в R продовжується до ендоморфізму в R. В роботі наведено деякі властивості таких кілець та отримано деякі цікаві результати. З використанням поняття AMGP-ін'єктивних кілець наведено деякі нові характеристики QF-кілець, напівпростих артінових кілець та простих артінових кілець. Institute of Mathematics, NAS of Ukraine 2013-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2527 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 11 (2013); 1476–1481 Український математичний журнал; Том 65 № 11 (2013); 1476–1481 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2527/1815 https://umj.imath.kiev.ua/index.php/umj/article/view/2527/1816 Copyright (c) 2013 Zhanmin Zhu |
| spellingShingle | Zhanmin, Zhu Чжанмін, Чжу Almost MGP-Injective Rings |
| title | Almost MGP-Injective Rings |
| title_alt | Майже MGP-iн'єктивнi кiльця |
| title_full | Almost MGP-Injective Rings |
| title_fullStr | Almost MGP-Injective Rings |
| title_full_unstemmed | Almost MGP-Injective Rings |
| title_short | Almost MGP-Injective Rings |
| title_sort | almost mgp-injective rings |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2527 |
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