Abstract
Let us call a ring R (without identity) to be right symmetric if for any triple a,b,c,∈R abc = 0 then acb = 0. Such rings are neither symmetric nor reversible (in general) but are semicommutative. With an idempotent they take care of the sheaf representation as obtained by Lambek. Klein 4-rings and their several generalizations and extensions are proved to be members of such class of rings. An extension obtained is a McCoy ring and its power series ring is also proved to be a McCoy ring.
Highlights
A ring R is symmetric if for any triple a1, a2, a3 ∈ R, a1a2a3 = 0 for any permutation σ : {1, 2,3} → {1, 2,3} aσ (1)aσ (2)aσ (3) = 0
The class of symmetric rings lie between the classes of reduced and reversible rings and they have been extensively studied and generalized in various directions, for instance, some references are [2]-[4], and [5]
Let us say that a ring R is right symmetric if for any triple a1, a2, a3 ∈ R, a1a2a3 = 0, a1a3a2 = 0
Summary
(2014) On Extensions of Right Symmetric Rings without Identity. K. Nauman given in [1], can be extended to right (or left) symmetric rings with idempotents (Proposition 2.7). ZC2 , it is reversible, and if it satisfies ZC3 , it is symmetric They proved that ZC3 implies ZCn , ∀n ≥ 3, but the converse need not be true in general ([7]; Example I-4). For a ring R with 1R , clearly, every symmetric ring is reversible, but the converse may not be true, for instance, see ([7]; Example 1-5). In ([8]: Example 7], Mark proved that the group ring Z2 (= Q8 ) : {xt : t ∈ Q8} where Q8 = {±1, ±i, ± j, ±k} is the group of quaternions, is reversible but not symmetric.
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