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$R$ doesn't need to be connected, so long as $k$ is (and if $R$ is connected then $k$ is, since a nontrivial idempotent of $k$ would be a nontrivial central idempotent of $R$). Also, $R$ doesn't need …

There are very easy examples (e.g., the path algebra of an $A_2$ quiver) where there is a nonzero projective module $P$ with $\operatorname{Hom}_A(P,\underline{A})=0$, so I assume you mean $\sup(\empt …

I'll give three answers, which basically say: (A) it doesn't matter, (B) it's not true, and (C) here's (a sketch of) a proof. But before that, there are a couple of relevant conditions in Linckelmann' …

Suppose $U$ has this property. Let $V=FG$ and let $W$ be the direct sum of $|G|$ copies of the trivial representation. Then $V\otimes U$ is free of rank $\dim(U)$, $W\otimes U$ is a direct sum of $|G| …

I don't know exactly the relationship between the two definitions, but they're not the same, as the first one depends only on the class in $\operatorname{Ext}^n_\Lambda(C,A)$ but the second doesn't. F …

A similar question was asked on math.stackexchange a few years ago, and I posted the following answer. I've just looked again at Conlon's paper, and I'm afraid it's a bit short on explicit examples. = …

I think I have a counterexample. Let $\operatorname{char}(k)=3$ and let $G$ be the symmetric group $S_{3}$. Then $kG$ has two simple modules: the trivial module $k$ and another one-dimensional module …

Here's an elementary proof that $2n-2$ is a lower bound. Suppose that $$V_1\xrightarrow{\alpha_1}V_2\xrightarrow{\alpha_2}\dots\xrightarrow{\alpha_{n-2}}V_{n-1}\xrightarrow{\alpha_{n-1}}V_n$$ is a rep …

At least if the field is algebraically closed (or sufficiently large), a finite group has a normal $p$-complement if and only if its principal block is local. For example, this is Corollary 6.13 of Na …

This is Theorem 30.29 in Curtis, Charles W.; Reiner, Irving, Methods of representation theory, with applications to finite groups and orders. Vol. I, Pure and Applied Mathematics. A Wiley-Interscience …

Yes. Let $M$ be any $A$-module. Then its socle is a direct sum of simple modules: $\operatorname{soc}A=\bigoplus_iS_i$. $A$ is a finite dimensional algebra, so the dual $\mathrm{Hom}_k(A,k)$ of $A$ is …

Assuming that the question is about finitely generated modules, I think that the following gives a finite dimensional Frobenius algebra $A$ that is a counterexample. In fact, for any non-projective fi …

Even if $A$ is a finite dimensional $k$-algebra, there may be no bound on the dimension of $\text{End}_A(X)/m$. Let $Q$ be the Kronecker quiver (i.e., the quiver with two vertices and two arrows from …

Let $Y$ be the module in the second row between $N$ and $A$. Using the mesh relations, the composition $$\tau^{-1}X\to N\to C\to M\to X$$ can be rewritten (up to a sign) as $$\tau^{-1}X\to N\to Y\to A …

$\operatorname{Hom}_{\mathcal{T}}(\mathcal{M}, \mathcal{M}[>0]) = 0$ means that $\operatorname{Hom}_{\mathcal{T}}\left(X, \Sigma^i(Y)\right) = 0$ for all objects $X,Y$ of $\mathcal{M}$ and all integer …