A Matrix Type Of Prime Avoidance Lemma

As a simple but very useful lemma in commutative algebra, the prime avoidance lemma may state as follows:Let P1,P2,…, Ps be prime ideals of a commutative ring R. If I is an ideal of R such that I is not contained in any Pi, thenThe main purpose of this paper is to prove a matrix type of prime avoidance lemma and give an application of the result to the finite free resolution of a module with projective dimension 2. The main results of the paper are the following two theorems.Theoreml Let A be a kr × r matrix with coefficients in a commutative ring R. Let P1, P2,..., Ps be prime ideals and J an ideal of R. Assume that the rank ideal I of A does not lie in J and any Pi,1≤i≤s. Then there exists a sequence of elementary row transformations of matrices which keep the first (k-1)r rows of A unchange and transform A to B such that the last r × r minor of B does not lie in J ∪ P1 ∪ P2 ∪... ∪ Ps.Theorem2 Let R be a noetherian ring, and A, B, C, D be r × r matrices with coefficients in R. Set Let I, J be the rank ideals of M, N, respectively. Assume that (ⅰ) r(M)=r(N)=r; (ⅱ) CA+DB=0; (ⅲ) depth I= 2.We have (1) If depth J=1, the J=aI; (2) if depth J=2, then I=J.