# Properties inherited from $R$ by Laurent polynomials $R[x;x^{-1}]$

I wonder if there is a paper about the conditions going up to Laurent Polynomial rings

For example the Laurent polynomial preserves the condition of reversibility of ring R

For a ring $R$ and the polynomial ring $R[x]$

When $R[x]$ is a reversible ring, so is the Laurent polynomial ring $R[x;x^{-1}]$.

I found it also preserves a few other conditions such as Symmetric, and Armendariz, but I can't find any which generalized the preserving property of $R[x;x^{-1}]$.

Particularly I want to know whether this will hold when $R[x]$ is reduced.

If you have any idea or information regarding this, please let me know.

• Commutative rings or in general? You mean the Laurent polynomials inhereting the reduced condition? The commutative case is easy: when $R[x]$ is reduced, so are its localizations, and $R[x;x^{-1}]$ is a localization of $R[x]$. – rschwieb Feb 25 '13 at 14:30

Many results about properties preserved by polynomial, power and Laurent series constructions are special cases of more general results about semigroup rings, so this is often the best place to look for such results. For example, for the mentioned Armendariz property see the following paper: Marks; Mazurek; Ziembowski: A unified approach to various generalizations of Armendariz rings, 2010, which studies the Armendariz condition in generalized power series ring $\rm\,R[[S,\omega]],\,$ where $\rm\,R\,$ is any ring, $\rm\,S\,$ is a strictly ordered monoid, and $\rm\,\omega: S\to End(R)\,$ is a monoid homomorphism, which is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Malcev–Neumann Laurent series rings.
One can find many classical results in Gilmer's Commutative Semigroup Rings, e.g. the following result on principal and Dedekind semigroup rings (the Laurent case is $\rm\,S = \Bbb Z\,$ below).
Theorem $\ \$ TFAE for a semigroup ring R[S], with unitary ring R, and nonzero torsion-free cancellative monoid S.
1) $\$ R[S] is a PIR (Principal Ideal Ring)
2) $\$ R[S] is a general ZPI-ring (i.e. a Dedekind ring, see below)
3) $\$ R[S] is a multiplication ring (i.e. $\rm\ I \supset\ J \Rightarrow\ I\ |\ J\$ for ideals $\rm\:I,J\:$)
4) $\$ R is a finite direct sum of fields, and S is isomorphic to $\mathbb Z$ or $\mathbb N$
A general ZPI-ring is a ring theoretic analog of a Dedekind domain i.e. a ring where every ideal is a finite product of prime ideals. A unitary ring R is a general ZPI-ring $\iff$ R is a finite direct sum of Dedekind domains and special primary rings (aka SPIR = special PIR) i.e. local PIRs with nilpotent max ideals. ZPI comes from the German phrase "Zerlegung in Primideale" = factorization in prime ideals. The classical results on Dedekind domains were extended to rings with zero divisors by S. Mori circa 1940, then later by K. Asano and, more recently, by R. Gilmer. See Gilmer's book "Commutative Semigroup Rings" sections 18 (and section 13 for the domain case).