A UNIFIED APPROACH TO VARIOUS GENERALIZATIONS OF ARMENDARIZ RINGS

作者：Marks Greg^*; Mazurek Ryszard; Ziembowski Michal

来源：Bulletin of the Australian Mathematical Society, 2010, 81(3): 361-397.

DOI：10.1017/S0004972709001178

• skew generalized power series ring
• (S, omega)-Armendariz
• semicommutative
• 2-primal
• reversible
• reduced
• uniserial

Let R be a ring, S a strictly ordered monoid, and omega:S -> End(R) a monoid homomorphism. The skew generalized power series ring R[[S, omega]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev-Neumann Laurent series rings. We study the (S, omega)-Armendariz condition on R, a generalization of the standard Armendariz condition from polynomials to skew generalized power series. We resolve the structure of (S, omega)Armendariz rings and obtain various necessary or sufficient conditions for a ring to be (S, omega)-Armendariz, unifying and generalizing a number of known Armendariz-like conditions in the aforementioned special cases. As particular cases of our general results we obtain several new theorems on the Armendariz condition; for example, left uniserial rings are Armendariz. We also characterize when a skew generalized power series ring is reduced or semicommutative, and we obtain partial characterizations for it to be reversible or 2-primal.

• 出版日期2010-6