caa a22 4500 378883771 CHVBK 20180305123438.0 cr unu---uuuuu 161128e20040101xx s 000 0 eng 10.1515/156939204774148820 doi (NATIONALLICENCE)gruyter-10.1515/156939204774148820 Gorbatov E.V. Standard basis of a polynomial ideal over commutative Artinian chain ring [Elektronische Daten] [E.V. Gorbatov] We construct a standard basis of an ideal of the polynomial ring R[X] = R[x 1, . . . , 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 χ of a polynomial ideal over the field = R/ rad(R) can be lifted to a standard basis of the same cardinality over R with respect to the natural epimorphism ν : R[X] → [X] if and only if there is an ideal I R[X] such that I is a free R-module and Ī = (χ). Copyright 2004, Walter de Gruyter Discrete Mathematics and Applications Walter de Gruyter 14/1(2004-01-01), 75-101 0924-9265 14:1<75 2004 14 dma https://doi.org/10.1515/156939204774148820 text/html Onlinezugriff via DOI 1 research article jats NATIONALLICENCE 856 40 https://doi.org/10.1515/156939204774148820 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Gorbatov E.V. NATIONALLICENCE 773 0- Discrete Mathematics and Applications Walter de Gruyter 14/1(2004-01-01), 75-101 0924-9265 14:1<75 2004 14 dma CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-gruyter