Fiber-complemented graphs II. Retractions and endomorphisms

On zero-divisor graphs of quotient rings and complemented zero-divisor graphs

For an arbitrary ring $R$, the zero-divisor graph of $R$, denoted by $Gamma (R)$, is an undirected simple graph that its vertices are all nonzero zero-divisors of $R$ in which any two vertices $x$ and $y$ are adjacent if and only if either $xy=0$ or $yx=0$. It is well-known that for any commutative ring $R$, $Gamma (R) cong Gamma (T(R))$ where $T(R)$ is the (total) quotient ring of $R$. In this...

Endomorphisms of undirected modifications of directed graphs

If z is an automorphism (or an endomorphism) of H, then it is also an automorphism (or an endomorphism) of %H and of %H. Hence the automorphism group Aut H is a subgroup of Aut 9?H and of Aut %H. Is there any other relation between the groups Aut H, Aut CBH and Aut %H? More in detail, which spans of groups, see Fig. 1, where m,, m2 are monomorphisms, can be realized by directed graphs H (in the...

on zero-divisor graphs of quotient rings and complemented zero-divisor graphs

for an arbitrary ring $r$, the zero-divisor graph of $r$, denoted by $gamma (r)$, is an undirected simple graph that its vertices are all nonzero zero-divisors of $r$ in which any two vertices $x$ and $y$ are adjacent if and only if either $xy=0$ or $yx=0$. it is well-known that for any commutative ring $r$, $gamma (r) cong gamma (t(r))$ where $t(r)$ is the (total) quotient ring of $r$. in this...

Endomorphisms and synchronization, 2: Graphs and transformation monoids

Some Endomorphisms of Ii1 Factors: Part Ii

For any finite dimensional C-algebra A with a trace vector ~s whose entries are rational numbers, we give an endomorphism Φ of the hyperfinite II1 factor R such that: ∀ k ∈ N, Φ(R) ∩R = ⊗A. The canonical trace τ on R extends the trace vector ~s on A. Therefore the minimal projection is not necessarily equivalent to each other.