Prime and semiprime submodules of Rn and a related Nullstellensatz for Mn(R)

Abstract

Let [Formula: see text] be a commutative ring with [Formula: see text] and [Formula: see text] a natural number. We say that a submodule [Formula: see text] of [Formula: see text] is semiprime if for every [Formula: see text] such that [Formula: see text] for [Formula: see text] we have [Formula: see text]. Our main result is that every semiprime submodule of [Formula: see text] is equal to the intersection of all prime submodules containing it. It follows that every semiprime left ideal of [Formula: see text] is equal to the intersection of all prime left ideals that contain it. For [Formula: see text] where [Formula: see text] is an algebraically closed field we can rephrase this result as a Nullstellensatz for [Formula: see text]: For every [Formula: see text], [Formula: see text] belongs to the smallest semiprime left ideal of [Formula: see text] that contains [Formula: see text] iff for every [Formula: see text] and [Formula: see text] such that [Formula: see text] we have [Formula: see text].