论文标题
在集体戒指的连接
On the joins of group rings
论文作者
论文摘要
给定一个集合$ \ {g_i \} _ {i = 1}^d $的有限组和一个环$ r $，我们定义了环$ m_n（r）$（r）$（$ n = \ sum_ {i = \ sum_ {i = 1}^d | g_i |）$，该$ compasses concompasses compassess $ ring $ r [g_i $ r [g_i] $ r [g_i] $ r [g_i] $ - 矩阵。该环的确切定义是灵感来自图形理论中的构造，称为“共同的图”联合。我们称此戒指为组戒指的连接，并用$ \ Mathcal {j} _ {g_1，\ dots，g_d}（r）$表示。在本文中，我们介绍了$ \ Mathcal {J} _ {g_1，\ dots，g_d}（r）$的代数结构的系统研究。我们证明它具有环形结构，并表征其中心，一组单位和雅各布森激进分子。当$ r = k $是一个代数封闭的字段时，我们将在$ \ nathcal {j} _ {g_1，\ dots，g_d}（k）$上得出一个不可约模块数的公式。我们还展示了傅立叶变换的块伸展如何提供循环对角线定理的概括，以与循环矩阵结合在一起，并在联接代数与其wedderburn组件之间显式同构。
Given a collection $\{ G_i\}_{i=1}^d$ of finite groups and a ring $R$, we define a subring of the ring $M_n(R)$ ($n = \sum_{i=1}^d|G_i|)$ that encompasses all the individual group rings $R[G_i]$ along the diagonal blocks as $G_i$-circulant matrices. The precise definition of this ring was inspired by a construction in graph theory known as the joined union of graphs. We call this ring the join of group rings and denote it by $\mathcal{J}_{G_1,\dots, G_d}(R)$. In this paper, we present a systematic study of the algebraic structure of $\mathcal{J}_{G_1,\dots, G_d}(R)$. We show that it has a ring structure and characterize its center, group of units, and Jacobson radical. When $R=k$ is an algebraically closed field, we derive a formula for the number of irreducible modules over $\mathcal{J}_{G_1,\dots, G_d}(k)$. We also show how a blockwise extension of the Fourier transform provides both a generalization of the Circulant Diagonalization Theorem to joins of circulant matrices and an explicit isomorphism between the join algebra and its Wedderburn components.
