Null Ideals of Matrices

ABSTRACT

Let R denote a commutative ring with identity. Let A ∈ M _n×n (R). The kernel of the natural map ϑ _A  : R[x] → M _n×n (R) given by ϑ _A (a _t x ^t  + ··· + a ₁ x + a ₀) = a _t A ^t  + ··· + a ₁ A + a ₀ I _n is denoted by N _A and called the null ideal of A. In this article, the question of when N _A is a principal ideal in R[x] is studied. For n = 2, N _A is principal for every matrix A ∈ M _{2 × 2} (R) if and only if R is a P.P. ring, i.e., every principal ideal in R is projective. If R contains only finitely many minimal primes, then N _A is principal for every A ∈ M _2×2(R) if and only if R is a finite product of integral domains. If R contains only finitely many minimal primes, then N _A is principal for all n ≥ 1 and for all A ∈ M _n×n (R) if and only if R is a finite product of normal domains. For fixed n ≥ 1 and A ∈ M _n×n (R), conditions on various R-submodules of N _A are given which imply N _A is principal.

Key Words:

• Matrices over commutative rings
• Null ideals

Mathematics Subject Classification:

• Primary 13B22, 15A21
• Secondary 13B25, 12E05, 11C08, 11C20

ACKNOWLEDGMENT

I wish to thank the referee of this article for Theorem 1.2. This theorem allowed me to shorten a couple of proofs in the original version of this article.

Communicated by I. Swanson.