∗-Regularity in the ring of matrices over the ring of integers modulo 𝑛

Abstract

For any positive integer 𝑛 ≥ 2, we give necessary and sufficient conditions of the existence of the Moore-Penrose inverse of any square matrix over the ring of integers modulo 𝑛. In particular, the formula for the Moore-Penrose inverse of any 2 × 2 matrix is also explained if it exists. We also characterize all values of 𝑘 and 𝑛 for which the ring of all 𝑘 × 𝑘 matrices over the ring of integers modulo 𝑛 is ∗-regular with respect to the matrix transposition as an involution. It turns out that the ring of 𝑘 × 𝑘 matrices over the ring of integers modulo 𝑛 is ∗-regular if and only if 𝑛 is square-free and either 𝑘 = 1 or 𝑘 = 2 and each prime divisor of 𝑛 must have the form 4𝑚 + 3 for some nonnegative integer 𝑚.

Keywords

• moore-penrose inverse
• ∗-regular ring
• regular number