\(\partial\)-operations and \(H_V\)-fields.
The paper is devoted to the study of \(\partial\)-operations, initiated by the author [in Ratio Math. No. 15, 25-36 (2005)]. Let \((R,+,\cdot)\) be a ring and \(f,g\colon R\to R\) are mappings. The author defines the hyperoperations \((\partial_+)\) and \((\partial_\cdot)\), that are called `theta-operations', on \(R\) by \[ x\partial_+y=\{f(x)+y,x+f(y)\},\quad x\partial_\cdot y=\{g(x)y,xg(y)\} \] for all \(x,y\in R\). It is proved that \((R,\partial_+,\partial_\cdot)\) is an \(H_v\)-near-ring. Also, some other properties of this ring are investigated. In the last section of paper, examples are given from the ring of integers and it is shown that under certain conditions the resulting \(H_v\)-near-ring is an \(H_v\)-field.
scientific article; zbMATH DE number 5323446