Left and right distributive rings (Q1916662)

Distributive rings and modules are studied and the relations between distributivity and diverse properties of rings (such as the properties to be invariant, localizable, reduced etc.) are exposed. We list short summaries of the main results. The right distributive right antisingular rings are described in different ways, in particular as rings \(A\) such that for each maximal right ideal \(M\) the right ring of quotients \(A_M\) exists and is a right chain domain. The right distributivity of a reduced ring \(A\) with the condition: (*) for every \(a\in A\) there is a positive integer \(n\) such that \(a^n A=Aa^n\), implies the left distributivity of \(A\). If \(A\) is a right distributive ring algebraic over its center, then the quotient ring \(R=A/N\) is a left distributive ring with condition (*) and the prime radical \(N\) of \(R\) coincides with the set of all nilpotent elements of \(A\). Each right Bezout module over a right quasiinvariant ring is distributive. If \(A\) is a right Bezout ring in which all maximal right ideals are ideals, then \(A\) is right distributive and in this case each of the following conditions implies the left distributivity of \(A\): (1) \(A\) is algebraic over its center and semiprime; (2) \(A\) is a left Bezout ring.