Further results on skew Hurwitz series ring (I)
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Further results on skew Hurwitz series ring (I) Kamal Paykan1 Received: 3 July 2019 / Accepted: 7 November 2019 / Published online: 15 November 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract In this paper, we continue the study of skew Hurwitz series ring (H R, α), where R is a ring equipped with an endomorphism α. In particular, we investigate the problem when a skew Hurwitz series series ring (H R, α) has the same Goldie rank as the ring R, and we obtain partial characterizations for it to be serial semiprime. Finally, we will obtain criterion for skew Hurwitz series rings to be right non-singular. Keywords Skew Hurwitz series ring · Goldie rank · Serial semiprime ring · Non-singular Mathematics Subject Classification 16S99 · 16W60 · 16S36 · 16N40
1 Introduction Rings of formal power series have been of interest and have had important applications in many areas, one of which has been differential algebra. In an earlier paper by Keigher [7], the ring of Hurwitz series, a variant of the ring of formal power series was considered, and some of its properties, especially its categorical properties, were studied. In the papers [8,9] Keigher demonstrated that the ring of Hurwitz series has many interesting applications in differential algebra and in the discussion about weak normalization. Its product, a product of sequences using binomial coefficients, was studied in papers by Fleiss [3] and Taft [26]. While there are many studies of these rings over a commutative ring, very little is known about them over a noncommutative ring. Ring-theoretical properties of skew Hurwitz series rings have been investigated by many authors (see [7–9,15,18,20,21] and [27]). Throughout this paper, R denotes an associative ring with unity and α : R → R is an endomorphism such that α(1) = 1. The ring (H R, α) of skew Hurwitz series over a ring R is defined as follows: the elements of (H R, α) are functions f : N → R, where N is the set of integers greater or equal than zero. The operation of addition in (H R, α) is componentwise and the operation of multiplication is defined, for every f , g ∈ (H R, α), by: n n f g(n) = f (k)α k (g(n − k)) for each n ∈ N, k k=0
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Kamal Paykan [email protected] Department of Basic Sciences, Garmsar Branch, Islamic Azad University, Garmsar, Iran
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n where is the binomial coefficient defined for all n, k ∈ N with n ≥ k by n!/(k!(n −k)!). k It is easy to verify that with the pointwise addition and the multiplication (H R, α) becomes a ring which we will call the skew Hurwitz series over a ring R. In the case where the endomorphism α is the identity, nwe denote H R instead of (H R, α). If one identifies a skew formal power series ∞ n=0 an x ∈ R[[x; α]] with the function f such that f (n) = an , then multiplication in (H R, α) is similar to the usual product of skew formal power series, except that binomial coefficients appear in each term in the product introduced above. A ring R is called right finite Goldie rank (known also as Goldie dimension
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