Basic properties of Hurwitz series rings

Chain conditions on composite Hurwitz series rings

“…In [2], Keigher called such a ring the ring of Hurwitz series and examined its ring theoretic properties. Since then, many works on the ring of Hurwitz series have been done ( [3][4][5]). …”

“…In [4], Benhissi and Koja studied when the Hurwitz rings H.R/ and h.R/ are Noetherian rings, S-Noetherian rings or satisfy the ascending chain condition on principal ideals. They showed that if char.R/ D 0, then H.R/ is a Noetherian ring if and only if h.R/ is a Noetherian ring, if and only if R a Noetherian ring containing Q [4, Corollary 7.7].…”

“…They showed that if char.R/ D 0, then H.R/ is a Noetherian ring if and only if h.R/ is a Noetherian ring, if and only if R a Noetherian ring containing Q [4, Corollary 7.7]. Also, they proved that for an anti-Archimedean subset S of R with zero characteristic containing an element s 0 2 S divisible in R by all the nonzero positive integers, if R is an S-Noetherian ring, then h.R/ is an S-Noetherian ring [4, Theorem 9.4]; and if R is an S -Noetherian ring and S consists of nonzerodivisors, then H.R/ is an S-Noetherian ring [4,Theorem 9.6].…”

“…Note that H.R/ (resp., h.R/) is not isomorphic to ROEOEX (resp., ROEX ) because R does not contain Q. Since R D R n f0g is anti-Archimedean subset of R and R is R -Noetherian, it follows from [4, Theorem 9.6] (resp., [4,Theorem 9.4]) that H.R/ (resp., h.R/) is an R -Noetherian ring. (3) Let R Â D be an extension of integral domains, where R is defined in .2/.…”

“…It was shown in [18] that R is présimplifiable if and only if Z.R/ Â 1 C U.R/. In [4], the authors studied when Hurwitz rings H.R/ and h.R/ are présimplifiable. In this section, we modify some properties of elements (units and nilpotent) of H.R/ and h.R/ in [4] to give equivalent conditions for composite Hurwitz series rings and composite Hurwitz polynomial rings to be présimplifiable.…”