Operators of discrete convolution type and their Noetherianess

The class of all multiplication operators is described which up to a compact summand commute with the convolution operators in $L_p(H)$ $(1\le p\le\infty)$, where $H$ is an infinite discrete Abelian group. A criterion is established for the operators in the algebra generated by these multiplication operators and the convolution operators to be Noetherian.