AN ANDERSON'S THEOREM ON NONCOMMUTATIVE RINGS

Abstract

Abstract. Let R be a ring and I be a proper ideal of R . For the caseof R being commutative, Anderson proved that ( ∗ ) there are only finitelymany prime ideals minimal over I whenever every prime ideal minimalover I is finitely generated. We in this note extend the class of ringsthat satisfies the condition ( ∗ ) to noncommutative rings, so called ho-momorphically IFP , which is a generalization of commutative rings. Asa corollary we obtain that there are only finitely many minimal primeideals in the polynomial ring over R when every minimal prime ideal ofa homomorphically IFP ring R is finitely generated. Throughout every ring is associative with identity unless otherwise stated.The n by n matrix ring over a ring R is denoted by Mat n ( R ). Due to Bell [2], aright (or left) ideal I of a ring R is said to have the insertion-of-factors-property (simply IFP ) if ab ∈ I implies aRb ⊆ I for a,b ∈ R . A ring R is called IFP ifthe zero ideal of R has the IFP. For a ring R and an ideal I , note that