Convolution rings of multiplications of an abelian group (Q1187325)

Let \((A,+)\) be a non-trivial abelian group, and let \((L,+)\) be the group of left distributive multiplications of \((A,+)\). If \(X\) is a fixed finite subset of \(A\), then \((L,+,\bullet)\) is a ring, where \(f \bullet g(a,b) = \sum_{x \in X} f(x,g(a,b))\) for \(f\), \(g \in L\). This paper investigates the structure of this ring (ideals, left identities, chain conditions) and of some related modules and rings. In particular, it is shown that \((L,+)\) is not a nil group.