The standard basis of a polynomial ideal over a commutative Artinian chain ring

We construct a standard basis of an ideal of the polynomial ring $R[X]=R[x_1,\ldots,x_k]$ over commutative Artinian chain ring $R$, which generalises a Gröbner base of a polynomial ideal over fields. We adopt the notion of the leading term of a polynomial suggested by D. A. Mikhailov and A. A. Nechaev, but using the simplification schemes introduced by V. N. Latyshev. We prove that any canonical generating system constructed by D. A. Mikhailov and A. A. Nechaev is a standard basis of the special form. We give an algorithm (based on the notion of $S$-polynomial) which constructs standard bases and canonical generating systems of an ideal. We define minimal and reduced standard bases and give their characterisations. We prove that a Gröbner base $\chi$ of a polynomial ideal over the field $\bar R=R/\operatorname{rad}(R)$ can be lifted to a standard basis of the same cardinality over $R$ with respect to the natural epimorphism $\nu\colon R[X]\to \bar R[X]$ if and only if there is an ideal $I\triangleleft R[X]$ such that $I$ is a free $R$-module and $\bar{I}=(\chi)$.
The research was supported by the Russian Foundation for Basic Research, grant 02-01-00218, and by the President of the Russian Federation program of support of leading scientific schools, grant 1910.2003.1.