On prime left ideals in \(\Gamma\)-rings (Q1585269)

Let \(M\) be a \(\Gamma\)-ring, and let \(g\) be a mapping that assigns to each \(m\in M\) an ideal \(g(m)\) of \(M\) such that for all \(m,n\in M\) it holds that (i) \(m\in g(m)\); (ii) \(n\in g(m)+A\) where \(A\) is an ideal of \(M\) implies \(g(n)\subseteq g(M)+A\). Let \(\gamma\in\Gamma\). A left ideal \(P\) of \(M\) is called left \(g^1\)-prime (resp. left \(g^1\)-\(\gamma\)-prime) if for all \(m,n\in M\), \(g(m)\Gamma n\subseteq P\) (resp. \(g(m)\gamma n\subseteq P\)) implies \(m\in P\) or \(n\in P\). Left \(g^1\)-systems and left \(g^1\)-\(\gamma\)-systems are defined in the obvious way which ensures that a left ideal \(P\) is left \(g^1\)-prime (resp. left \(g^1\)-\(\gamma\)-prime) if and only if the complement of \(P\) is a left \(g^1\)-system (resp. \(g^1\)-\(\gamma\)-system). Clearly every left \(g^1\)-\(\gamma\)-prime left ideal is left \(g^1\)-prime. An example is given to show that the converse does not hold in general. Various elementary results are proved about \(g^1\)-\(\gamma\)-prime left ideals. The intersection of all the left \(g^1\)-prime left ideals (resp. left \(g^1\)-\(\gamma\)-prime left ideals) of \(M\) is called the left \(g^1\)-prime radical (resp. left \(g^1\)-\(\gamma\)-prime radical) of \(M\) and is denoted \(P_{g^1}(M)\) (resp. \(P_{g^1\gamma}(M)\)). It is shown that \(P_{g^1\gamma}(M)=\{m\in M\mid\) every left \(g^1\)-\(\gamma\)-system which contains \(m\) contains \(0\}\).