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Units on the Gauss extension of a Galois ring. (English) Zbl 1342.13010

Let \(p\) be prime and let n and r be positive integers. Denote by \(Z\) the ring of integers and by \(\mathrm{GR}(p^n, r)\) the quotient ring \( Z_{p{^n}} [x]/\left\langle f(x)\right\rangle\), where \(f(x)\) is a monic, basic irreducible polynomial of degree \(r\) in \(Z_{p{^n}} [x]\). The ring \(\mathrm{GR}(p^n, r)\) is called a Galois extension of the local ring \(Z_{p{^n}}\). Let \(Z[i]\) be the ring of Gaussian integers i.e. \( Z[i]\) is an extension of \(Z\) by the imaginary unit \(i\). J. Cross gives a full description of the unit group of \(Z_n[i]\) for some special values of \(n\) [J. T. Cross, Am. Math. Mon. 90, 518–528 (1983; Zbl 0525.12001)]. G. Tang et al. completely determined the unit group of \(Z[i]\) for an arbitrary \(n\) [G. Tang et al., J. Guangxi Norm. Univ., Nat. Sci. 28, No. 2, 38–41 (2010; Zbl 1240.13027)]. In this article the authors study the Gauss extension \(\mathrm{GR}(p^n, r)[i] \cong \mathrm{GR}(p^n, r)/\left\langle x^2+1\right\rangle\). They determine the prime spectrum and the set of zero-divisors of \(\mathrm{GR}(p^n, r)[i] =R\) (Theorem 2.3). The main result of the article is Theorem 3.4, in which the authors give a description, up to isomorphism, of the unit group of \(R\).

MSC:

13B05 Galois theory and commutative ring extensions
13M05 Structure of finite commutative rings
16U60 Units, groups of units (associative rings and algebras)
16L99 Local rings and generalizations
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