Distributive monoid algebras
Let \(AG\) be the monoid ring of a von Neumann regular monoid \(G\) over a field \(A\). The monoid ring \(AG\) is called right (left) distributive ring if the set of all right (left) ideals of \(AG\) is a distributive lattice. In this paper the author proves that the right (left) distributivity of \(AG\) is equivalent to the assertion that \(G\) is a strong semilattice of groups \(G_ j\), indexed by the elements of some semilattice, and for any pair \((A,G_ j)\) one of the following conditions is satisfied: a) \(G_ j\) is an abelian group with periodic part \(T_ j\), the group \(G_ j/T_ j\) is isomorphic to a subgroup of the additive group of rational numbers, and \(T_ j\) contains no element of order \(p\) if \(\text{char }A = p > 0\); b) \(G_ j\) is a periodic abelian group and \(G_ j\) contains no direct product of two groups of order \(p\) if \(\text{char }A = p > 0\); c) \(\text{char }A = 0\), \(G_ j\) is a Hamiltonian group, \(G_ j \cong K \times H_ j\), where \(K\) is the quaternion group and \(H_ j\) is a periodic abelian group, and if \(n\) is the order of an arbitrary element of \(H_ j\), then the equation \(u^ 2 + y^ 2 + z^ 2 = 0\) has only the zero solution in the ring \(A[x]/N\), \(N\) being the ideal of \(A[x]\) generated by the \(n\)-th cyclotomic polynomial. Moreover, the author proves an analogous theorem, when \(A\) is a commutative ring with identity.
scientific article; zbMATH DE number 484245
scientific article