Rings which are sums of finite fields (Q1284214)

It is shown that a semigroup ring \(R[S]\) is a direct sum of finite fields if and only if \(R\) is a ring of this type, \(S\) is a commutative semigroup in which every principal ideal is finite and \(S\) is torsion-disjoint with \(R\) (that is, \(s^n=t^n\) for \(s,t\in S\), \(s\neq t\), implies that \(R\) has no \(n\)-torsion). A characterization of rings that are direct sums of finite fields is also obtained for a certain class of rings graded by a semilattice. Finally, if \(\mathcal K\) is the class of rings that are sums of finite fields and a finite commutative ring \(R\) is a sum of its subrings \(R_i\in{\mathcal K}\), \(i=1,\dots,n\), then \(R\in{\mathcal K}\).