On a Banach algebra of entire functions with a weighted Hadamard multiplication

New algebraic-analytic properties of a previously studied Banach algebra $\mathcal{A}(\mathbf{p})$ of entire functions are established. For a given fixed sequence $(\mathbf{p}(\mathbf{n}){)}_{\mathbf{n}\ge \mathbf{0}}$ of positive real numbers, such that ${\lim }_{n\to \mathrm{\infty }}\mathbf{p}(n{)}^{\frac{1}{n}}=\mathrm{\infty }$, the Banach algebra $\mathcal{A}(\mathbf{p})$ is the set of all entire functions $f$ such that $f(z)={\sum }_{n=0}^{\mathrm{\infty }}\widehat{f}(n)z^n$ ($z\in \mathbb{C}$), where the sequence $(\widehat{f}(n){)}_{n\ge 0}$ of Taylor coefficients of $f$ satisfies $\widehat{f}(n)=O(\mathbf{p}(n{)}^{-1})$ for $n\to \mathrm{\infty }$, with pointwise addition and scalar multiplication, a weighted Hadamard multiplication $\ast$ with weight given by $\mathbf{p}$ (i.e., $(f\ast g)(z)={\sum }_{n=0}^{\mathrm{\infty }}\mathbf{p}(n)\widehat{f}(n)\widehat{g}(n)z^n$ for all $z\in \mathbb{C}$), and the norm $\mathrm{\parallel }f\mathrm{\parallel }={\sup }_{n\ge 0}\mathbf{p}(n)\mathrm{\mid }\widehat{f}(n)\mathrm{\mid }$. The following results are shown: The Bass and the topological stable ranks of $\mathcal{A}(\mathbf{p})$ are both $1$. $\mathcal{A}(\mathbf{p})$ is a Hermite ring, but not a projective-free ring. Idempotents and exponentials in $\mathcal{A}(\mathbf{p})$ are described, and it is shown that every invertible element of $\mathcal{A}(\mathbf{p})$ has a logarithm. A generalised necessary and sufficient `corona-type condition’ on the matricial data $(A,b)$ with entries from $\mathcal{A}(\mathbf{p})$ is given for the solvability of $Ax=b$ with $x$ also having entries from $\mathcal{A}(\mathbf{p})$. The Krull dimension of $\mathcal{A}(\mathbf{p})$ is infinite. $\mathcal{A}(\mathbf{p})$ is neither Artinian nor Noetherian, but is coherent. The special linear group over $\mathcal{A}(\mathbf{p})$ is generated by elementary matrices.