On the joins of group rings

Given a collection $\{G_i{\}}_{i=1}^d$ of finite groups and a ring $R$, we define a subring of the ring $M_n(R)$ ($n={\sum }_{i=1}^d\mathrm{\mid }{G}_i\mathrm{\mid })$ 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.