# Noetherianity and Specht problem for varieties of bicommutative algebras

## TL;DR

This paper proves that finitely generated bicommutative algebras are weakly noetherian and provides a positive solution to the Specht problem for their varieties over any field, advancing understanding of their algebraic structure.

## Contribution

It establishes weak noetherianity for finitely generated bicommutative algebras and solves the Specht problem for their varieties over arbitrary fields.

## Key findings

- Finitely generated bicommutative algebras satisfy the ascending chain condition.
- The Specht problem has a positive solution for varieties of bicommutative algebras.
- Results hold over fields of any characteristic.

## Abstract

Nonassociative algebras satisfying the polynomial identities x(yz)=y(xz) and (xy)z=(xz)y are called bicommutative. We prove the following results: (i) Finitely generated bicommutative algebras are weakly noetherian, i.e., satisfy the ascending chain condition for two-sided ideals. (ii) We give the positive solution to the Specht problem (or the finite basis problem) for varieties of bicommutative algebras over an arbitrary field of any characteristic.