On 𝑃-Coherent Endomorphism Rings
A ring is called right 𝑃-coherent if every principal right ideal is finitely presented. Let $M_R$ be a right 𝑅-module. We study the 𝑃-coherence of the endomorphism ring 𝑆 of $M_R$. It is shown that 𝑆 is a right 𝑃-coherent ring if and only if every endomorphism of $M_R$ has a pseudokernel in add $M_R; S$ is a left 𝑃-coherent ring if and only if every endomorphism of $M_R$ has a pseudocokernel in add $M_R$. Some applications are given.