Order in a special class of rings and a structure theorem

Abian, Alexander

doi:10.1090/S0002-9939-1975-0374222-8

Order in a special class of rings and a structure theorem
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Proc. Amer. Math. Soc. 52 (1975), 45-49 Request permission

Addendum: Proc. Amer. Math. Soc. 61 (1976), 188.

Abstract:

Below a special class of not necessarily associative or commutative rings $A$ is considered which is characterized by the property that $A$ has no nonzero nilpotent element and that a product of elements of $A$ which is equal to zero remains equal to zero no matter how its factors are associated. It is shown that $(A, \leqslant )$ is a partially ordered set where $x \leqslant y$ if and only if $xy = {x^2}$. Also it is shown that $(A, \leqslant )$ is infinitely distributive, i.e., $r\sup {x_i} = \sup r{x_i}$. Finally, based on Zorn’s lemma it is shown that $A$ is isomorphic to a subdirect product of not necessarily associative or commutative rings without zero divisors.

References

Alexander Abian, Direct product decomposition of commutative semi-simple rings, Proc. Amer. Math. Soc. 24 (1970), 502–507. MR 258815, DOI 10.1090/S0002-9939-1970-0258815-X
W. D. Burgess and R. Raphael, Abian’s order relation and orthogonal completions for reduced rings, Pacific J. Math. 54 (1974), 55–64. MR 360708
M. Chacron, Direct product of division rings and a paper of Abian, Proc. Amer. Math. Soc. 29 (1971), 259–262. MR 274512, DOI 10.1090/S0002-9939-1971-0274512-X