ON THE DOT PRODUCT GRAPH OF A COMMUTATIVE RING II

Mohammad ABDULLA; Ayman BADAWI

doi:https://doi.org/10.24330/ieja.768135

Makale detayları

ON THE DOT PRODUCT GRAPH OF A COMMUTATIVE RING II

Yazarlar:

Mohammad ABDULLA | Ayman BADAWI

Yayıncı:

Abdullah HARMANCI

Ana Başlıklar:

Matematik

Dergi:

International Electronic Journal of Algebra 2020 Vol.28 Issue 28, pp.61-74

Belge Tipi:

Makale

Yayın Tarihi:

7/2020

Belge Dili:

İngilizce

Document ID (Org No.):

2000717868

Sıklık:

Aylık

eISSN:

1306-6048

pISSN:

1306-6048

DOI:

https://doi.org/10.24330/ieja.768135

Özet:

 

In 2015, the second-named author introduced the dot product
graph associated to a commutative ring AAA. Let AAA be a
commutative ring with nonzero identity, 1≤n<∞1≤n<∞1 \leq  n  < \infty  be
an integer,  and R=A×A×⋯×AR=A×A×⋯×AR = A \times A \times \cdots \times A (nnn
times).  We recall that the total dot product graph of  RRR
is the (undirected) graph TD(R)TD(R)TD(R) with vertices R∗=R∖{(0,0,…,0)}R∗=R∖{(0,0,…,0)}R^* = R\setminus
\{(0, 0, \dots, 0)\}, and two distinct vertices xxx and yyy are
adjacent if and only if x⋅y=0∈Ax⋅y=0∈Ax\cdot y = 0 \in A (where x⋅yx⋅yx\cdot y
denotes the normal dot product of xxx and yyy). Let Z(R)Z(R)Z(R) denote
the set of all zero-divisors of RRR. Then the zero-divisor
dot product graph of RRR is the induced subgraph ZD(R)ZD(R)ZD(R) of
TD(R)TD(R)TD(R)  with vertices Z(R)∗=Z(R)∖{(0,0,…,0)}Z(R)∗=Z(R)∖{(0,0,…,0)}Z(R)^* = Z(R) \setminus \{(0, 0, \dots,
0)\}. Let  U(R)U(R)U(R) denote the set of all units of RRR. Then the unit dot product graph of RRR is the induced subgraph
UD(R)UD(R)UD(R) of TD(R)TD(R)TD(R)  with vertices U(R)U(R)U(R). In this paper, we study
the structure  of TD(R)TD(R)TD(R), UD(R)UD(R)UD(R), and ZD(R)ZD(R)ZD(R) when A=ZnA=ZnA = Z_n or
A=GF(pn)A=GF(pn)A = GF(p^n), the finite field with pnpnp^n elements, where n≥2n≥2n \geq
2 and ppp is a prime positive integer.

Makale detayları

ON THE DOT PRODUCT GRAPH OF A COMMUTATIVE RING II

Yazarlar: Mohammad ABDULLA | Ayman BADAWI

Yayıncı: Abdullah HARMANCI

Ana Başlıklar: Matematik

Dergi: International Electronic Journal of Algebra 2020 Vol.28 Issue 28, pp.61-74

Belge Tipi: Makale

Yayın Tarihi: 7/2020

Belge Dili: İngilizce

Document ID (Org No.): 2000717868

Sıklık: Aylık

eISSN: 1306-6048

pISSN: 1306-6048

DOI: https://doi.org/10.24330/ieja.768135

Sayı

Yazarlar:

Mohammad ABDULLA | Ayman BADAWI

Yayıncı:

Abdullah HARMANCI

Ana Başlıklar:

Matematik

Dergi:

International Electronic Journal of Algebra 2020 Vol.28 Issue 28, pp.61-74

Belge Tipi:

Makale

Yayın Tarihi:

7/2020

Belge Dili:

İngilizce

Document ID (Org No.):

2000717868

Sıklık:

Aylık

eISSN:

1306-6048

pISSN:

1306-6048

DOI:

https://doi.org/10.24330/ieja.768135