STRONGLY GRADED RINGS WHICH ARE KRULL RINGS

Year 2019, , 120 - 128, 08.01.2019

İndah Emilia Wijayanti Hidetoshi Marubayashi İwan Ernanto Akira Ueda

https://doi.org/10.24330/ieja.504135

Cited By: 1

Abstract

Let $R = \oplus_{n \in \Z} R_{n}$ be a strongly graded ring of type $\Z$ and $R_{0}$ is a prime Goldie ring. It is shown that the following three conditions are equivalent: (i) $R_{0}$ is a $\Z$-invariant Krull ring, (ii) $R$ is a Krull ring and (iii) $R$ is a graded Krull ring. We completely describe all $v$-invertible $R$-ideals in $Q$, where $Q$ is a quotient ring of $R$.

 

Keywords

Strongly graded ring, prime Goldie ring, Krull ring, v-invertible R-ideal

References

• E. Akalan, H. Marubayashi and A. Ueda, Generalized hereditary Noetherian prime rings, J. Algebra Appl., 17(8) (2018), 1850153 (22 pp).
• H. Marubayashi, E. Nauwelaerts and F. Van Oystaeyen, Graded rings over arithmetical orders, Comm. Algebra, 12(5-6) (1984), 745-775.
• H. Marubayashi and F. Van Oystaeyen, Prime Divisors and Noncommutative Valuation Theory, Lecture Notes in Mathematics, 2059, Springer, Heidelberg, 2012.
• H. Marubayashi, S. Wahyuni, I. E. Wijayanti and I. Ernanto, Strongly graded rings which are maximal orders, Sci. Math. Jpn., 31 (2018), 2018-5.
• C. Nastasescu and F.Van Oystaeyen, Graded Ring Theory, North-Holland Mathematical Library, 28, North-Holland Publishing Co., Amsterdam-New York, 1982.
• B. Stenstrom, Rings of Quotients, An introduction to methods of ring theory, 217, Springer-Verlag, New York-Heidelberg, 1975.
• F. Van Oystaeyen and A. Verschoren, Relative Invariants of Rings, The commutative theory, Monographs and Textbooks in Pure and Applied Mathematics, 79, Marcel Dekker, Inc., New York, 1983.
• S. Wahyuni, H. Marubayashi, I. Ernanto and Sutopo, Strongly graded rings which are generalized Asano rings, preprint.