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On Some Generalizations of the Reversibility in Nonunital Rings
2019, Journal of The Korean Mathematical Society
https://doi.org/10.4134/JKMS.J180009
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Abstract
This paper is intended as a discussion of some generalizations of the notion of a reversible ring, which may be obtained by the restriction of the zero commutative property from the whole ring to some of its subsets. By the INCZ property we will mean the commutativity of idempotent elements of a ring with its nilpotent elements at zero, and by ICZ property we will mean the commutativity of idempotent elements of a ring at zero. We will prove that the INCZ property is equivalent to the abelianity even for nonunital rings. Thus the INCZ property implies the ICZ property. Under the assumption on the existence of unit, also the ICZ property implies the INCZ property. As we will see, in the case of nonunital rings, there are a few classes of rings separating the class of INCZ rings from the class of ICZ rings. We will prove that the classes of rings, that will be discussed in this note, are closed under extending to the rings of polynomials and formal power series. All rings considered in this paper are assumed to be associative but not necessarily with unit. The standard extension of a ring R to a unital ring with the help of the ring of integers is denoted by R 1 . The sets of idempotent elements in R and nilpotent elements in R are denoted by E(R) and N (R) respectively. J. Lambek in introduced the notion of a symmetric ring understood as a unital ring R in which rst = 0 implies rts = 0 for any r, s, t ∈ R, and proved that an equivalent condition on a unital ring R to be symmetric is that 0 for any positive integer n, any elements r 1 , r 2 , . . . , r n ∈ R and any permutation σ of the set 1, 2, . . . , n . D. D. Anderson and V. Camillo in [1] continued the study of rings whose zero
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- 2.7. For every ring R, the following statements are equivalent: 1. erf = ef r holds for any e, f ∈ E(R) and r ∈ R;
- etf = ef t holds for any e, f ∈ E(R) and t ∈ N (R);
- ere = er holds for any e ∈ E(R) and r ∈ R;
- te = 0 implies et = 0 for any e ∈ E(R) and t ∈ N (R);
- re = 0 implies eRr = 0 for any e ∈ E(R) and r ∈ R;
- te = 0 implies eRt = 0 for any e ∈ E(R) and t ∈ N (R).
- Proof. The implications 1 ⇒ 4 ⇒ 9 ⇒ 7 ⇒ 8, 1 ⇒ 2 ⇒ 5 ⇒ 10 ⇒ 8 and 1 ⇒ 3 ⇒ 6 are obvious. In the proofs of all three implications 6 ⇒ 4, 8 ⇒ 4 and 4 ⇒ 1, we let e, f ∈ E(R) and r ∈ R. Then, as we know, e+er-ere ∈ E(R) and er-ere ∈ N (R). In the case when the statement 6 holds, since e(e+er-ere)e = e(e + er -ere), it follows that ere = er. In the case when the statement 8 References
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