On graded coherent-like properties in trivial ring extensions. (English) Zbl 1497.13002
The authors introduce and study the notions of graded-\(v\)-coherent, graded-quasi-coherent and graded-finite conductor rings and their transfer in the graded trivial extension. Among others, the following results are proved:
- 1.
- Assume that the grading monoid \(G\) is a group. Let \(A\) be a graded integral domain which is not a field, \(A_{H}\) its homogeneous quotient field graded by \(G\) and let \(R:=A\ltimes A_{H}\) be the graded trivial extension of \(A\) by \(A_{H}\). Then \(R\) is a graded-\(v\)-coherent ring if and only if \(A\) is graded-\(v\)-coherent.
- 2.
- Assume that the grading monoid \(G\) is a group. Let \((A, M)\) be a graded local ring and \(E\) a graded \(A\)-module with \(ME=0\). Let \(R= A\ltimes E\) be the graded trivial extension. The following statements are equivalent.
- (i)
- \(R\) is a graded quasi-coherent (respectively, graded finite conductor) ring.
- (ii)
- \(A\) is a graded quasi-coherent (respectively, graded finite conductor) ring, \(M\) is finitely generated and \(E\) is an \(A/M\)-vector space of finite rank.
Reviewer: A. Mimouni (Dhahran)
Keywords:
graded trivial extension; trivial extension; graded modules and rings; coherent rings and modules; graded-\(v\)-coherenceReferences:
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