The Erdős–Burgess constant finiteness conjecture for commutative rings

Let $R$ be a commutative unitary ring. Write $J(R)$ for its Jacobson radical, and let $\textsc{I}(\mathcal{S}_R)$ denote the Erdős–Burgess constant of the multiplicative semigroup $\mathcal{S}_R$. Suppose that

$$R/J(R)\cong B\times\prod\limits_{i=1}^t\mathbb{F}_{q_i},$$

where $B$ is an infinite Boolean unitary ring, $t\geq 0$, the $\mathbb{F}_{q_i}$ are finite fields, and $J(R)$ is finite. Erdős–Burgess constant finiteness conjecture. Under these conditions, $\textsc{I}(\mathcal{S}_R)$ is finite. The preceding theorem gives these conditions as necessary for finiteness; the conjecture asserts their sufficiency and extends the established Noetherian and semi-local cases.