Demi-primal algebras
A finite algebra \({\mathfrak A}=<A,F>\) is called demi-primal if it has no proper subalgebra and every function \(f:A^ n\to A\) admitting the automorphisms of A is a polynomial of A. If \({\mathfrak M}\) is a faithful unitary right R-module, where R is a ring with 1, let \(T({\mathfrak M})\) be the set of operations \(x_ 0r_ 0+...+x_ kr_ k+m,\) with all \(r_ i\in R\), \(m\in M\), and \(r_ 0+...+r_ k=1.\) If q is a prime power, let \(K=GF(q).\) For \(n\geq 1\), let \(K_ n\) be the full \(n\times n\) matrix ring over K. Let \({\mathfrak N}_{n,q}\) be the faithful unitary right module over \(K_ n\), and let \({\mathfrak H}_{n,q}\) be the algebra \(<K^ n;T({\mathfrak N}_{n,q})>\). For prime p, let \({\mathfrak U}_ p\) be the unary algebra on K whose operations are the translations \(x_ 0+a\) (\(a\in A)\). The main result is: A finite algebra with transitive automorphism group is demi- primal if and only if it is simple, it has no proper subalgebra, and it is not equivalent to any of the algebras \({\mathfrak U}_ p\) or \({\mathfrak H}_{n,q}\), where p is prime, \(n\geq 1\), and q is a prime power.
scientific article; zbMATH DE number 3861228
scientific article