Communications in Algebra· 2013Q2
On the Annihilator Graph of a Commutative Ring
- 130citations
- Q2SCImago
- 2013year
Short summary
The annihilator graph AG(R) of a commutative ring R is shown to be connected with diameter at most two and girth at most four (if it has a cycle).
AI-generated from the title and abstract; the full text is not read.
Key points
- The annihilator graph AG(R) is defined on the non-zero zero-divisors of a commutative ring R.
- Two vertices x and y are adjacent in AG(R) if ann R (xy) ≠ ann R (x) ∪ ann R (y).
- AG(R) is connected with diameter at most two and girth at most four (if cyclic).
- For reduced rings, AG(R) = Γ(R) iff R has exactly two minimal prime ideals.
AI-generated from the title and abstract; the full text is not read.
Abstract
Let R be a commutative ring with nonzero identity, Z(R) be its set of zero-divisors, and if a ∈ Z(R), then let ann R (a) = {d ∈ R | da = 0}. The annihilator graph of R is the (undirected) graph AG(R) with vertices Z(R)* = Z(R)∖{0}, and two distinct vertices x and y are adjacent if and only if ann R (xy) ≠ ann R (x) ∪ ann R (y). It follows that each edge (path) of the zero-divisor graph Γ(R) is an edge (path) of AG(R). In this article, we study the graph AG(R). For a commutative ring R, we show that AG(R) is connected with diameter at most two and with girth at most four provided that AG(R) has a cycle. Among other things, for a reduced commutative ring R, we show that the annihilator graph AG(R) is identical to the zero-divisor graph Γ(R) if and only if R has exactly two minimal prime ideals.
The authors' abstract, as published at the source. Communications in Algebra, 2013 · DOI ↗
Continue with a free account
Ask the paper: 3 free questions a day about this paper; save it, get its citation, new summaries every day for your field. Takeaways are Premium.
Continue free on the webSign in with Google or Apple; no card needed. You come back to this paper.
On your phone:
Field: Algebra and Number Theory
Algebra and Number TheoryMathematics