Chain conditions in computable rings

Friedman, Simpson, and Smith showed that, over RCA$_0$, the statements ``Every ring has a maximal ideal'' and ``Every ring has a prime ideal'' are equivalent to ACA$_0$ and WKL$_0$, respectively. More recently, Downey, Lempp, and Mileti have shown that, over RCA$_0$, the statement ``Every ring that is not a field contains a nontrivial ideal'' is equivalent to WKL$_0$.

In this article we explore the reverse mathematical strength of the classic theorems from commutative algebra which say that every Artinian ring is Noetherian, and every Artinian ring is of finite length. In particular we show that, over RCA$_0$, the former implies WKL$_0$ and is implied by ACA$_0$, while over RCA$_0$+B$\Sigma_2$, the latter is equivalent to ACA$_0$.