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I have the following question: Let $A$ be an Artin algebra. Let $S_1$ and $S_2$ be simple modules in $\text{mod}(A)$ and let $P(S_1)$ be the projective cover of $S_1$. Let $f: P(S_1) \rightarrow S_2$ ...

How can I show the following: Let $f: M \rightarrow N$ be a morphism in $\text{mod}(A)$, where $A$ is an Artin algebra. Suppose $f \neq 0$. Then there exists a simple module $S$ with its injective ...

I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is a part of the paper: Definition: Let $f: X \rightarrow Y$ be an irreducible morphism ...

I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is Theorem 2.7: And this is part of it's proof, in which the direction (2) $\Rightarrow $ ...

I am reading a paper called A NOTE ON THE RADICAL OF A MODULE CATEGORY by CLAUDIA CHAIO AND SHIPING LIU. This is part of the proof of Lemma 2.3 $A$ is assumed to be an Artin algebra and mod(A) the ...

Do you know of any self-injective basic algebra $A$ over a field $k$ such that its enveloping algebra $A^{\mathrm{op}}\otimes_k A$ is not self-injective? The algebra $A$ cannot be finite-dimensional, ...