Gelfand-Kirillov dimension of bicommutative algebras

Abstract

We first offer a fast method for calculating the Gelfand-Kirillov dimension of a finitely presented commutative algebra by investigating certain finite set. Then we establish a Gröbner–Shirshov bases theory for bicommutative algebras, and show that every finitely generated bicommutative algebra has a finite Gröbner–Shirshov basis. As an application, we show that the Gelfand-Kirillov dimension of a finitely generated bicommutative algebra is a nonnegative integer.

Keywords:

• Commutative algebra
• bicommutative algebra
• Gelfand-Kirillov dimension
• Gröbner basis
• Gröbner–Shirshov basis

2020 Mathematics Subject Classifications:

• 13P10
• 16S15
• 16P90
• 17A30
• 17A61

Acknowledgments

The authors would like to thank the referee for his/her insightful comments and valuable suggestions. The authors are also grateful to L. A. Bokut and Yu Li for bringing the topic of bicommutative algebras to our attention.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Y.C. is supported by the NNSF of China (11571121, 12071156); Z.Z. is supported by the NNSF of China (12101248), by the China Postdoctoral Science Foundation 2021M691099 and by the Young Teacher Research Cultivation Foundation of South China Normal University 20KJ02.