Ideal structure of Hurwitz series rings

Ali Benhissi

Abstract: We study the ideals, in particular, the maximal spectrum and the set of idempotent elements, in rings of Hurwitz series.

Let $A$ be a commutative ring with identity. The elements of the ring $HA$ of Hurwitz series over $A$ are formal expressions of the type $\displaystyle f=\sum_{i=0}^{\infty}a_iX^i$ where $a_i\in A$ for all $i$. Addition is defined termwise. The product of $f$ by $\displaystyle g=\sum_{i=0}^{\infty}b_iX^i$ is defined by $\displaystyle f*g=\sum_{n=0}^{\infty}c_nX^n$ where $\displaystyle c_n=\sum_{k=0}^n (_k^n)a_kb_{n-k}$ and $(_k^n)$ is a binomial coefficient. Recently, many authors turned to this ring and discovered interesting applications in it. See for example [K] and [2]. The natural homomorphism $\epsilon:HA\longrightarrow A$, is defined by $\epsilon(f)=a_0$. \item[K] Keigher, W. F.: On the ring of Hurwitz series. Comm. Algebra {\bf 25}(6) (1997), 1845--1859. \item[L] Liu, Z.: Hermite and PS-rings of Hurwitz series. Comm. Algebra {\bf 28}(1) (2000), 299--305.

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