Geodesic
On noncommutative Gr\"obner bases over rings
Fundamentalʹnaâ i prikladnaâ matematika, Tome 10 (2004) no. 4, pp. 91-96

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Let $R$ be a commutative ring. It is proved that for verification whether a set of elements $\{f_\alpha\}$ of the free associative algebra over $R$ is a Gröbner basis (with respect to some admissible monomial order) of the (bilateral) ideal that the elements $f_\alpha $ generate it is sufficient to check reducibility to zero of $S$-polynomials with respect to $\{f_\alpha\}$ iff $R$ is an arithmetical ring. Some related open questions and examples are also discussed. 
@article{FPM_2004_10_4_a6,
     author = {E. S. Golod},
     title = {On noncommutative {Gr\"obner} bases over rings},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {91--96},
     publisher = {mathdoc},
     volume = {10},
     number = {4},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2004_10_4_a6/}
}
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E. S. Golod. On noncommutative Gr\"obner bases over rings. Fundamentalʹnaâ i prikladnaâ matematika, Tome 10 (2004) no. 4, pp. 91-96. http://geodesic.mathdoc.fr/item/FPM_2004_10_4_a6/