The relationships between graded ideals and subspaces

Zariski Topologies on Graded Ideals

“…The next proposition has been given in ( [8], Theorem 10), but we obtain it by combining Proposition 2.6 and Proposition 2.17.…”

“…In the rest of our paper, we study some algebraic and topological tools for graded ideals and some characterizations for graded rings. We begin with the following results that have been proved by Ortaç Önes and Mustafa Alkan in [8]. They will be needed in our study.…”

“…Remark 2.16. Proposition 2.15 has been proved in [8,Theorem 4], for any G-graded ring, where G is any group, but this is not true because their proof depended on saying that √ P is a graded ideal of R, and this is not true in general by [7, Exercises 17 and 13 on pp. 127-128].…”

“…127-128]. On the other hand, it has been proved in [2,Lemma 2.13], that if P is a graded ideal of a Z-graded ring R, then √ P is a graded ideal of R. So, Proposition 2.15 is a correction for [8,Theorem 4]. In fact, we do not know whether the general result in Proposition 2.15 is true for any group G, but there is a fault in the proof and the former proof works for Z-graded rings.…”