Rings Which Are Sums of Finite Fields

Abstract

We describe all semigroup rings and band-graded rings which are direct sums of finite fields. We show that the class of rings which are direct sums of finite fields is closed for taking sums of two subrings. Academic PressSemigroup rings of arithmetical semigroups have nice applications in analytic number theory (see, for example, [14, Sect. 2.1]). We shall describe all semigroup rings which are direct sums of finite fields (Theorem 3). To this end we consider another important construction (Theorem 2) and establish an interesting ring-theoretic property of the class of rings which are direct sums of finite fields (Theorem 1).The preservation of ring properties by sums of rings has been actively investigated by many authors. Let K be a class of rings, and let a ring R be a sum of its subrings R G , where i"1, 2 , n. Suppose that all the R G are in K. Does it follow that R belongs to K? This question has been considered for several classes K; see [1,2,6,8] for references. We only mention that a longstanding difficult problem in this area has been recently solved in [9] and [12]. Namely, examples of rings which are not nil but are sums of two locally nilpotent subrings were constructed. Taking a homomorphic image of a ring introduced in [9], a simpler version of the example was given in [12]. Moreover, it was shown that there exists a primitive ring which is a sum of two Wedderburn radical subrings, which answered several questions known 89