结构矩阵环上的整数值多项式
Integer-valued polynomials over structural matrix rings
- Graz University of Technology（格拉茨工业大学）
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AI总结：
该研究探讨结构矩阵环上的整数值多项式与零多项式，证明两类多项式的相关代数性质，并基于标量系数多项式刻画带矩阵系数的对应多项式，推广了全矩阵环与上三角矩阵环的相关定理。
AI中文摘要：
我们研究结构矩阵环上的整数值多项式与零多项式，这类环的元素为矩阵，在固定预序下，仅当行索引早于列索引时，矩阵元素才可非零。我们证明：以整环为元素的结构矩阵环上的整数值多项式构成一个环；以任意交换环为元素的结构矩阵环上的零多项式构成一个双边理想。在两种情形下，我们均根据标量系数多项式，给出了对应带矩阵系数多项式的刻画。这些结果推广了全矩阵环与上三角矩阵环的对应定理。
英文摘要：
We study integer-valued polynomials and null polynomials over structural matrix rings, that is, rings whose elements are matrices in which an entry may be nonzero only when its row index precedes its column index in a fixed preorder. We prove that the integer-valued polynomials over a structural matrix ring with entries in an integral domain form a ring, and that the null polynomials over a structural matrix ring with entries in an arbitrary commutative ring form a two-sided ideal. In both settings, we give a characterization of the corresponding polynomials with matrix coefficients in terms of scalar-coefficient polynomials. These results extend corresponding theorems for full matrix rings and upper triangular matrix rings.
