Given a finite simple graph $\Gamma$, the graph group $G \Gamma$ is the group with generators in one-to-one correspondence with the vertices of $\Gamma$ and with relations in one-to-one correspondence with the edges of $\Gamma$: two generators commute if and only if their associated vertices are adjacent in $\Gamma$. Let $\phi \colon G \Gamma \to \mathbb Z$ be a homomorphism which maps each generator to 0 or 1. We derive an explicit presentation for $\ker \phi$, and give a condition, dependent on $\Gamma$ and $\phi$, which guarantees the finite presentation of $\ker \phi$.

In this paper we give a construction of infinite monoids generated by the integer solutions of Euler's Diophantine equation $x^2 + y^2 = z^n$, $n \ge 2$.

Beginning with the observation that to every non-equilateral, isosceles triangle, there is a unique, noncongruent isosceles triangle with the same perimeter and the same area, we determine all such pairs with integer sides and integer areas. We also determine those pairs in which the sides in each isosceles triangle are relatively prime. Finally, we give an infinite family of pairs of triangles equal in perimeter and in area, one isosceles and one non-isosceles.

All rings are commutative rings with identity. $\tilde R$ denotes the set of all units in a ring $R$ together with 0 and it is clear that $R \backslash \tilde R = \emptyset$ if and only if $R$ is a field. In addition to some other results, it is shown that $R$ is not stable if and only if there exists a unimodular sequence $(y,u)$ in $R$ with $y \in R$ and $u \in R \backslash \tilde R$ such that $u$ is not a side divisor of $y$. For each $s \ge 1$, a sequence $a_1, a_2, \ldots , a_s , a_{s+1}$ of elements in a ring $R$ is said to be stable, whenever the ideal is $(a_1, a_2, \ldots , a_s , a_{s+1} ) = (a_1 + b_1 a_{s+1}, \ldots , a_s + b_s a_{s+1} )$ for some $b_1, b_2, \ldots , b_s$ in $R$. A sequence $a_1, a_2, \ldots , a_s , a_{s+1}$ of elements in $R$ is called a unimodular sequence provided that $(a_1, a_2, \ldots , a_s , a_{s+1} ) = R$. For any fixed positive integer $n$, we shall say $R$ is $n$-stable (simply, stable for $n=1$), whenever, for all $s \ge n$ any unimodular sequence, $a_1, a_2, \ldots , a_s , a_{s+1}$ in $R$ is stable. $u \in R \backslash \tilde R$ is said to be a side divisor of $y \in R$, if $u \vert y-z$ for some $z \in \tilde R$. Besides two other different proofs, we apply the above result to show that $R [ X ]$ is not stable for any commutative ring $R$. At the end, it is shown that any Artinian ring is stable.