Subalgebras that are cyclic as submodules

Abstract

Let R be an associative, commutative, unital ring. By a R-algebra we mean a unital R-module A together with a R-module homomorphism μ: ⊗ ⁿ_R A→A (n≥2). We raise the question whether such an algebra possesses either an idempotent or a nilpotent element. In section 1 an affirmative answer is obtained in case R=k is an algebraically closed field and dim_kA<∞, as well as in case R=ℝ, dim_ℝS<∞, and n≡0(2). Section 2 deals with the case of reduced rings R and R-algebras which are finitely generated and projective as R-modules. In section 3 we show that the “generic” algebra over an integral domain D fails to have nilpotent elements in any integral domain extending its base ring D_n,m, and thus acquires an idempotent element in some integral domain extending D_n,m.