Let R be a commutative ring with $Z(R)$ its set of zero-divisors. In this paper, we study the total graph of $R$, denoted by $T(Gamma(R))$. It is the (undirected) graph with all elements of R as vertices, and for distinct $x, yin R$, the vertices $x$ and $y$ are adjacent if and only if $x + yinZ(R)$. We study the chromatic number and edge connectivity of this graph.

Proposition 1. Let R be a ring and M an R-module. Then EndR(M), the set of R-linear maps from M to M , is a subring of End(M). Proof. Recall from 3.1.6 that the ring (End(M), +, ◦) is defined by (α + β)(m) = α(m) + β(m), (α ◦ β)(m) = α(β(m)). We must verify the four subring conditions from 3.2.2. (i) The additive identity of End(M) is z : M → M, m → 0M . Note that for any m,n ∈ M z(m + n) = 0M ...

Let R be a ring with center Z, Jacobson radical J , and set N of all nilpotent elements. Call R generalized periodic-like if for all x ∈ R \ (N ∪ J ∪ Z) there exist positive integers m, n of opposite parity for which xm − xn ∈ N ∩ Z. We identify some basic properties of such rings and prove some results on commutativity. Let R be a ring; and let N = N(R), Z = Z(R) and J = J(R) denote respective...

Let $R=\mathbb{Z}_4+u\mathbb{Z}_4,$ where $\mathbb{Z}_4$ denotes the ring of integers modulo $4$ and $u^2=0$. In the present paper, we introduce a new Gray map from $R^n$ to $\mathbb{Z}_{4}^{2n}.$ We study $(1+2u)$-constacyclic codes over $R$ of odd lengths with the help of cyclic codes over $R$. It is proved that the Gray image of $(1+2u)$-constacyclic codes of length $n$ over $R$ are cyclic c...

Introduction. Consider the region ^ *= {(x, y9 z)e R 3 \x + y ^ / (z) , 0 ^ z g L) obtained by rotating the curve f(z), 0 ^ x ^ L, about the z-axis. Denote by q(t, r, r') the fundamental solution of the heat equation in ^ with zero boundary value on d99 and let p(t9 r, V) = (l/(2ftf) ) * exp(-|r r'\/2t). Express everything in terms of cylindrical coordinates (p, 0, z). The purpose of this note ...

Proof of Proposition 1. With a slight abuse of notation, let us denote by D d the set of multivariate stochastic processes such that the aggregate demand equals d. Accordingly, Problem (5) can be decomposed as follows: ρ(y) = max d∈P max D∈D d max z∈F R(z, D) − R(y, D) = max d∈P max z∈F max D∈D d R(z, D) + max D∈D d −R(y, D). Let ξ j be the realized sales under a booking policy z when the deman...

Two lens spaces are given to show that Ohtsuki’s τ for rational homology spheres does not determine Kirby-Melvin’s {τ ′ r , r odd ≥ 3} By using partial Kirby-Melvin’s quautum SO(3) invariants {τ ′ r(M), r odd prime > |H1(M,Z)|} [KM], Ohtsuki [O] defined a topological invariant τ(M) = ∞ ∑ n=0 λn(M)(t − 1) ∈ Q[[t− 1]] for rational homology 3-sphere M . R.Lawrence [La] conjectured that λn(M) ∈ { Z...