Idempotents and Quasi-monic Polynomials

Abstract

In this paper, the definitions of quasi-orthogonal idempotent sequences and Iq-dimensions of a ring R are given. The relations between Iq-dimensions and block decomposition numbers of a ring are discussed. As a generalization of monic polynomials, the concept of quasi-monic polynomials over a ring is introduced. It is shown that, for a quasi-monic polynomial over a ring, the division algorithm holds. Suslin Lemma and Horrocks' Theorem are extended to the setting of quasi-monic polynomials. For a commutative ring R, if f(x) is a quasi-monic polynomial in R[x], then GD(R) = GD(R[x]/f(x)) is proved, where GD(R) denotes the global dimension of R.

Key Words:

• Idempotent
• Quasi-monic polynomial
• Dimension
• Horrocks' Theorem
• Division algorithm

Acknowledgments

The author would like to give many thanks to the referee for many useful suggestions and comments, which have greatly improved this paper.

This study was supported by the National Natural Science Foundation of China(10171011).