Ideals of the associative algebra operad

TL;DR

This paper establishes a correspondence between operadic ideals of the associative algebra operad and T-ideals, proving that the operad is noetherian and characterizing its ideals.

Contribution

It introduces a novel correspondence between operadic ideals and T-ideals, demonstrating the noetherian property of the associative algebra operad.

Findings

Operadic ideals of $ ext{As}$ correspond one-to-one with T-ideals.

The operad $ ext{As}$ is noetherian.

Every proper operadic ideal of $ ext{As}$ is singly generated.

Abstract

We prove a one-to-one correspondence between the operadic ideals of the operad $\As$ and $T$-ideals. As a consequence, we show that $\As$ is noetherian and that every proper operadic ideal of $\ias$ is generated by a single element.