The Direct Summand Conjecture

Let $R \rightarrow S$ be an injective ring homomorphism, where $R$ is a regular local ring and $S$ is module-finite over $R$. Direct Summand Conjecture. Then $R$ is a direct summand of $S$ as an $R$-module. This conjecture concerns the splitting of finite extensions of regular local rings and is one of the central homological conjectures in commutative algebra. Its resolution status is not specified in the supplied source.