论文标题
在交换有限链环上的一些特殊矩阵的决定因素
Determinants of some Special Matrices over Commutative Finite Chain Rings
论文作者
论文摘要
由于其良好的代数结构和广泛的应用，因此在有限场和有限有限链环上进行的循环矩阵引起了人们的关注。在许多情况下，此类矩阵上的矩阵与对角矩阵的延伸环有着封闭的联系。在本文中，研究了对角线和循环矩阵的决定因素，而有限链环$ r $带有残留场$ \ mathbb {f} _q $的决定因素。确定所有元素$ a $ in $ {r} $的$ {r} $上的$ n \ times n $ n $对角矩阵的数量，以及所有正整数$ n $。随后，对所有单位$ a $ a $ in $ {r} $的$ {r} $ cripculant矩阵的非字词$ n \ times n $循环矩阵的枚举和所有正整数$ n $均给出。在某些情况下，用固定决定因素的$ {r} $循环矩阵的单数$ n \ times n $循环矩阵的数量是通过循环矩阵和对角线矩阵之间的链接确定的。作为应用，给出了关于对角线和循环矩阵对交换有限的理想环的简要讨论。最后，发布了一些开放问题和猜想
Circulant matrices over finite fields and over commutative finite chain rings have been of interest due to their nice algebraic structures and wide applications. In many cases, such matrices over rings have a closed connection with diagonal matrices over their extension rings. In this paper, the determinants of diagonal and circulant matrices over commutative finite chain rings $R$ with residue field $\mathbb{F}_q$ are studied. The number of $n\times n$ diagonal matrices over ${R}$ of determinant $a$ is determined for all elements $a$ in $ {R}$ and for all positive integers $n$. Subsequently, the enumeration of nonsingular $n\times n$ circulant matrices over ${R}$ of determinant $a$ is given for all units $a$ in $ {R}$ and all positive integers $n$ such that $\gcd(n,q)=1$. In some cases, the number of singular $n\times n$ circulant matrices over ${R}$ with a fixed determinant is determined through the link between the rings of circulant matrices and diagonal matrices. As applications, a brief discussion on the determinant of diagonal and circulant matrices over commutative finite principal ideal rings is given. Finally, some open problems and conjectures are posted
