The topological nearring on the Euclidean plane which has a left identity which is not a right identity (Q1281622)

Main Theorem: Let \((\mathbb{R}^2,+,\circ)\) be a topological nearring where \(+\) is the usual addition on \(\mathbb{R}^2\) and suppose \((\mathbb{R}^2,+,\circ)\) has a left multiplicative identity which is not a right multiplicative identity. Then \((\mathbb{R}^2,+,\circ)\) is isomorphic to \((\mathbb{R}^2,+,*)\) where the binary operation \(*\) is given by \(v*w=v_1w\) and consequently, \((\mathbb{R}^2,+,\circ)\) is a ring.