On Algebraic Shift Equivalence of Matrices over Polynomial Rings

Details On Algebraic Shift Equivalence of Matrices over Polynomial Rings On Algebraic Shift Equivalence of Matrices over Polynomial Rings

2007-10-19

arxiv.org/abs/0710.3746v3

Mathematics

Rings and Algebras

8 pages

Scientific paper

The paper studies algebraic strong shift equivalence of matrices over $n$-variable polynomial rings over a principal ideal domain $D$($n\leq 2$). It is proved that in the case $n=1$, every non-zero matrix over $D[x]$ has a full rank factorization and every non-nilpotent matrix over $D[x]$ is algebraically strong shift equivalent to a nonsingular matrix. In the case $n=2$, an example of non-nilpotent matrix over $\mathbb{R}[x,y,z]=\mathbb{R}[x][y,z]$, which can not be algebraically shift equivalent to a nonsingular matrix, is given.

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