Groups

Abstract

Let R be not necessarily commutative ring, M a left R-module. We recall (see Introduction 3.1) that M is a simple (irreducible) R-module if any R-submodule either coincides with M or is equal to the zero submodule. The simple modules over the ring Z of integers are the cyclic groups of prime order p. The simple modules over the ring of polynomials F[X] of one variable X over an algebraically closed field F are one-dimensional. If the field is not algebraically closed then every module of this kind has the form of quotient module F[X]/(p(X)), where the polynomial p(X) is irreducible in F[X]. The structure of simple modules over an arbitrary principal ideal ring is similar. Finally, if R = F[X ₁,..., X _n ], then, according to the Hilbert Nullstellensatz, any irreducible module over R is finite-dimensional. Indeed an irreducible module is generated by any of its non-zero elements, so it is cyclic and, therefore, is representable in the form R/P, where P is some maximal ideal. Further, as was mentioned already, R/P is a field which is finitely generated over F as an algebra. From the Hilbert Nullstellensatz follows that each element from R/P is algebraic over F, i.e. R/P is a finitely generated algebraic extension of a field. Finally, R/P is finite-dimensional over F as was required. If F is algebraically closed then any irreducible R-module is one-dimensional.