Principal ideals in matrix rings

Abstract

Let R be a ring with a unity 1, and let n be a positive integer. It is well-known [3, p. 37]1 that every two-sided ideal of R" (the complete matrix ring of order n over R) is necessarily of the form M", where M is a two-sided ideal of R. Simple examples show that this result no longer holds for one-sided ideals. In this note we investigate the left ideals of R" in the case when R is a principal ideal ring (an integral domain in which every ideal is principal). We shall prove THEOREM 1: IfR is a principal ideal ring, then every left ,ideal ofRn is principal_ The proof of Theorem 1 depends upon the fact that if A is any p X q matrix over R, then a unit matrix V of Rp exists such that the p X q matrix VA is upper triangular [2 , p. 32]_ We also establish the following partial converse to Theorem 1: THEOREM 2: If R is not Noetherian or if R is a Dedekind ring but not a principal ideal ring, then Rn contains a nonprincipalleft ideal_ For general information on rings , see [3]. For information on Dedekind rings, see [1 , p. 101].