On structures of gamma-seminear-rings (Q2720364)

A seminear-ring is a triple \((R,+,\cdot)\) such that \((R,+)\) and \((R,\cdot)\) are semigroups and \((x+y)\cdot z=x\cdot z+y\cdot z\) for all \(x,y,z\in R\). A \(\Gamma\)-seminear-ring is a triple \((R,+,\Gamma)\) such that \((R,+,\gamma)\) is a seminear-ring for each \(\gamma\in\Gamma\) and \((x\gamma y)\mu z=x\gamma(y\mu z)\) for all \(x,y,z\in R\) and \(\gamma,\mu\in\Gamma\). In this paper a congruence relation \(\rho\) is defined on \((R,+,\Gamma)\), which enables a quotient \(\Gamma\)-seminear-ring \((R/\rho,+,\Gamma)\) to be defined. This enables a version of the fundamental homomorphism theorem to be proved for \(\Gamma\)-seminear-rings.