Let $\Lambda^r_n$ be the path algebra of linearly oriented quiver type $\mathbb{A}$ with $n$ vertices modulo $r$-th power radical, and let $\widetilde{\Lambda}^r_n$ cyclically $\widetilde{\mathbb{A}}$ radical. Adachi gave a recurrence relation for number $\tau$-tilting modules over $\Lambda^r_n$. In this paper, we show that same also holds $\widetilde{\Lambda}^r_n$. As an application, give new proof result by Asai on formulae support

Let $B$ be the one-point extension algebra of $A$ by an $A$-module $M$. We proved that every ICE-closed subcategory in$\mod A$ can extended to some subcategories B$.In same way, epibrick in $\mod epibricks B$.The number B$ and are denoted respectively as $m$, $n$.We conclude following inequality:$$m \geq 2n$$ This is analogical epibricks.As application, we get wide $\tau$-tilting modules $A$.

We define integrals for functions on finite-dimensional algebras, adapting methods from Leinster's research. This paper discusses the relationships between of defined subsets $\mathbb{I}_1 \subseteq {\mathit{\Lambda}}_1$ and $\mathbb{I}_2 {\mathit{\Lambda}}_2$ two under influence a mapping $\omega$, which can be an injection or bijection. explore four specific cases: $\bullet$ $\omega$ as monotone non-decreasing right-continuous function; injective, absolutely continuous bijection; identity...

Let $\Lambda,\Gamma$ be rings and $R=\left(\begin{array}{cc}\Lambda & 0 \\ M \Gamma\end{array}\right)$ the triangular matrix ring with $M$ a $(\Gamma,\Lambda)$-bimodule. $X$ right $\Lambda$-module $Y$ $\Gamma$-module. We prove that $(X, 0)$$\oplus$$(Y\otimes_\Gamma M, Y)$ is silting $R$-module if only both $X_{\Lambda}$ $Y_{\Gamma}$ are modules $Y\otimes_\Gamma M$ generated by $X$. Furthermore, we $\Lambda$ $\Gamma$ finite dimensional algebras over an algebraically closed field finitely...

Let $\Gamma $ be a split extension of finite-dimensional algebra $\Lambda by nilpotent bimodule $_\Lambda E_\Lambda $, and let $(T,P)$ pair in $\operatorname{mod} \Lambda with $P$ projective. We prove that $(T\otimes _\Lambda \Gamma _\Gamma ,

Let [Formula: see text] be the Auslander algebra of a finite-dimensional basic connected Nakayama with radical cube zero and simple modules. Then cardinality set consisting isomorphism classes tilting text]-modules is

Let B be the one-point extension algebra of A by an A-module X. We proved that every support τ-tilting can extended to a B-module two different ways. As consequence, it is shown there inequality |sτ‐tilt B|⩾2|sτ‐tilt A|.

Let $\Gamma$ be a split extension of finite-dimensional algebra $\Lambda$ by nilpotent bimodule $_\Lambda E_\Lambda$, and let $(T,P)$ pair in $\mod\Lambda$ with $P$ projective. We prove that $(T\otimes_\Lambda \Gamma_\Gamma, P\otimes_\Lambda \Gamma_\Gamma)$ is support $\tau$-tilting $\mod \Gamma$ if only \Lambda$ $\Hom_\Lambda(T\otimes_\Lambda E,\tau T_\Lambda)=0=\Hom_\Lambda(P,T\otimes_\Lambda E)$. As applications, we obtain necessary sufficient condition such for cluster-tilted...

It is well known that the relation-extensions of tilted algebras are cluster-tilted algebras. In this paper, we extend result to silted and prove some extension

It is well known that the relation extensions of tilted algebras are cluster-tilted algebras. In this paper, we extend result to silted and prove some extension

Let $\Lambda$ be an algebra with a indecomposable projective-injective module. Adachi gave method to construct the Hasse quiver of support $\tau$-tilting $\Lambda$-modules. In this paper, we will show that it can restricted modules.

Let $B$ be the one-point extension algebra of $A$ by an $A$-module $X$. We proved that every support $\tau$-tilting can extended to a $B$-module two different ways. As consequence, it is shown there inequality $$|\stilt B|\geqslant 2|\stilt A|.$$

Let [Formula: see text] be an algebra with indecomposable projective-injective module. Adachi gave a method to construct the Hasse quiver of support text]-tilting text]-modules. In this paper, we will show that it can restricted modules.

Let $\Lambda _n$ be a radical square zero Nakayama algebra with $n$ simple modules and $\Gamma the Auslander of _n$. We calculate number $|\tau \text {-tilt}\,\Gamma _n|$ $\tau $-tilting $|{\rm s}\tau \te