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Register a new userNonsplit module extensions over the one-sided inverse of k[x]
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Let $R$ be the associative $k$-algebra generated by two elements $x$ and $y$ with defining relation $yx=1$. A complete description of simple modules over $R$ is obtained by using the results of Irving and Gerritzen. We examine the short exact sequence $0rightarrow Urightarrow E rightarrow Vrightarrow 0$, where $U$ and $V$ are simple $R$-modules. It shows that nonsplit extension only occurs when both $U$ and $V$ are one-dimensional, or, under certain condition, $U$ is infinite-dimensional and $V$ is one-dimensional.
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An square matrix is $k$-Toeplitz if its diagonals are periodic sequences of period $k$. We find rational formulas for the determinant, the characteristic polynomial, and the elements of the inverse of a tridiagonal $k$-Toeplitz matrix (in particular, of any tridiagonal matrix) over any commutative unital ring, using only elementary linear algebra.
In this paper, we continue our study of the tensor product structure of category $mathcal W$ of weight modules over the Hopf-Ore extensions $kG(chi^{-1}, a, 0)$ of group algebras $kG$, where $k$ is an algebraically closed field of characteristic zero. We first describe the tensor product decomposition rules for all indecomposable weight modules under the assumption that the orders of $chi$ and $chi(a)$ are different. Then we describe the Green ring $r(mathcal W)$ of the tensor category $mathcal W$. It is shown that $r(mathcal W)$ is isomorphic to the polynomial algebra over the group ring $mathbb{Z}hat{G}$ in one variable when $|chi(a)|=|chi|=infty$, and that $r(mathcal W)$ is isomorphic to the quotient ring of the polynomial algebra over the group ring $mathbb{Z}hat{G}$ in two variables modulo a principle ideal when $|chi(a)|<|chi|=infty$. When $|chi(a)|le|chi|<infty$, $r(mathcal W)$ is isomorphic to the quotient ring of a skew group ring $mathbb{Z}[X]sharphat{G}$ modulo some ideal, where $mathbb{Z}[X]$ is a polynomial algebra over $mathbb{Z}$ in infinitely many variables.
We consider categories over a field $k$ in order to prove that smash extensions and Galois coverings with respect to a finite group coincide up to Morita equivalence of $k$-categories. For this purpose we describe processes providing Morita equivalences called contraction and expansion. We prove that composition of these processes provides any Morita equivalence, a result which is related with the karoubianisation (or idempotent completion) and additivisation of a $k$-category.
The ring operations and the metric on $C(X)$ are extended to the set $mathbb{H}_{nf}(X)$ of all nearly finite Hausdorff continuous interval valued functions and it is shown that $mathbb{H}_{nf}(X)$ is both rationally and topologically complete. Hence, the rings of quotients of $C(X)$ as well as their metric completions are represented as rings of Hausdorff continuous functions.
This paper addresses the development of analytical tools for the computation of the moments of random Gram matrices with one side correlation. Such a question is mainly driven by applications in signal processing and wireless communications wherein such matrices naturally arise. In particular, we derive closed-form expressions for the inverse moments and show that the obtained results can help approximate several performance metrics such as the average estimation error corresponding to the Best Linear Unbiased Estimator (BLUE) and the Linear Minimum Mean Square Error LMMSE or also other loss functions used to measure the accuracy of covariance matrix estimates.
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