Morphic property of a quotient ring over polynomial ring. (Q2872983)

In this short paper, the authors study morphic property of the quotient ring \(S_n\) of a polynomial ring \(R[x_1,\ldots,x_n]\) defined by \(S_n=R[x_1,\ldots,x_n]/(\{x_ix_j\mid i,j=1,\ldots,n\})\) over a ring \(R\). After deriving some elementary properties of the ring \(S_n\), the authors, in their main theorem, prove that if \(R\) is a strongly regular ring then morphic elements of the ring \(S_n\) are precisely elements of \(S_n\) of the form \(ue\) for a unit \(u\) in \(S_n\) and an idempotent \(e\) in \(R\). They prove that the ring \(S_n\) can never be a morphic ring. The authors conclude the paper by giving an example to show that in their theorem the condition ``strongly regularity of \(R\)'' cannot be replaced by ``unit regularity of \(R\)''.