Varieties and sums of rings (Q1580062)

The author studies varieties \(V\) of commutative algebras over a field, in which the sum of any two subalgebras of any algebra \(A\in V\) is a subalgebra of \(A\). Theorem: Let \(F\) be a field of characteristic \(p\) and let \(V\) be a nontrivial variety of commutative algebras over \(F\). Then \(V\) is closed under sums of subalgebras iff \(p>0\) and \(V\) is the product of \(N_{p^k}\) for some \(k\) and a variety \(S\) that is generated by a finite set of finite fields. \(N_t\) is the variety of all commutative algebras satisfying the identity \(x^t=0\).