Semigroups and rings whose zero products commute

Anderson, D. D.; Camillo, Victor

doi:10.1080/00927879908826596

Cited by 87 publications

(68 citation statements)

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“…Then the ring R = A/J is not right IIP but left IIP by a similar computation to (1). ( As we see in Example 2.3, one-sided IIP rings need not be Abelian when they do not have identity.…”

Section: Proof (1)⇒(2) (2)⇒(4) (1)⇒(3) and (3)⇒(5) Are Obviousmentioning

confidence: 90%

“…Due to Lambek [17], a ring R is called symmetric if rst = 0 implies rts = 0 for all r, s, t ∈ R. Anderson-Camillo [1] used the term ZC 3 for symmetric rings. Lambek proved, in [17,Proposition 1], that a ring R is symmetric if and only if r σ(1) r σ(2) · · · r σ(n) = 0 for any permutation σ of the set {1, 2, .…”

Section: Proof (1)⇒(2) (2)⇒(4) (1)⇒(3) and (3)⇒(5) Are Obviousmentioning

confidence: 99%

“…We will use this result freely. According to Cohn [6], a ring R is called reversible if ab = 0 implies ba = 0 for a, b ∈ R. Anderson-Camillo [1] …”

Section: Proof (1)⇒(2) (2)⇒(4) (1)⇒(3) and (3)⇒(5) Are Obviousmentioning

confidence: 99%

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Structure of Idempotents in Rings Without Identity

Kim¹,

Lee²,

Seo³

2014

Journal of the Korean Mathematical Society

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Abstract. We study the structure of idempotents in polynomial rings, power series rings, concentrating in the case of rings without identity. In the procedure we introduce right Insertion-of-Idempotents-Property (simply, right IIP) and right Idempotent-Reversible (simply, right IR) as generalizations of Abelian rings. It is proved that these two ring properties pass to power series rings and polynomial rings. It is also shown that π-regular rings are strongly π-regular when they are right IIP or right IR. Next the noncommutative right IR rings, right IIP rings, and Abelian rings of minimal order are completely determined up to isomorphism. These results lead to methods to construct such kinds of noncommutative rings appropriate for the situations occurred naturally in studying standard ring theoretic properties.

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“…Then the ring R = A/J is not right IIP but left IIP by a similar computation to (1). ( As we see in Example 2.3, one-sided IIP rings need not be Abelian when they do not have identity.…”

Section: Proof (1)⇒(2) (2)⇒(4) (1)⇒(3) and (3)⇒(5) Are Obviousmentioning

confidence: 90%

Section: Proof (1)⇒(2) (2)⇒(4) (1)⇒(3) and (3)⇒(5) Are Obviousmentioning

confidence: 99%

See 1 more Smart Citation

Structure of Idempotents in Rings Without Identity

Kim¹,

Lee²,

Seo³

2014

Journal of the Korean Mathematical Society

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“…(1) By Lemma 2.4(1, 2) every minimal non-commutative ring is isomorphic to the 2 × 2 upper triangular matrix ring over GF (2). So by Theorem 2.2 and Example 2.3(1, 2), R 3 over GF (2) is a minimal non-commutative strongly AB ring that is neither strongly right nor strongly left bounded.…”

Section: Theorem 22 a Ring R Is Strongly Right (Resp Left) Ab If Amentioning

confidence: 99%

“…Let R be a reduced ring, and consider R 3 over R; R 3 is IFP by [23, Proposition 1.2], but it is not strongly right bounded by the computation in (2).…”

Section: Theorem 22 a Ring R Is Strongly Right (Resp Left) Ab If Amentioning

confidence: 99%

On Rings Whose Right Annihilators Are Bounded

2009

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Abstract. Jacobson said a a right ideal would be called bounded if it contained a non-zero ideal, and Faith said a ring would be called strongly right bounded if every non-zero right ideal were bounded. In this paper we introduce a condition that is a generalisation of strongly bounded rings and insertion-of-factors-property (IFP) rings, calling a ring strongly right AB if every non-zero right annihilator is bounded. We first observe the structure of strongly right AB rings by analysing minimal non-commutative strongly right AB rings up to isomorphism. We study properties of strongly right AB rings, finding conditions for strongly right AB rings to be reduced or strongly right bounded. Relating to Ramamurthi's question (i.e. Are right and left SF rings von Neumann regular?), we show that a ring is strongly regular if and only if it is strongly right AB and right SF, from which we may generalise several known results. We also construct more examples of strongly right AB rings and counterexamples to several naturally raised situations in the process.2000 Mathematics Subject Classification. 16D25, 16D40, 16E50.

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Ring endomorphisms satisfying the central reversible property

Bhattacharjee

Chakraborty

2020

Proc Math Sci

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Semigroups and rings whose zero products commute

Cited by 87 publications

References 1 publication

Structure of Idempotents in Rings Without Identity

Structure of Idempotents in Rings Without Identity

On Rings Whose Right Annihilators Are Bounded

Ring endomorphisms satisfying the central reversible property

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