An Algorithm for Determining Whether the Quotient Ring Modulo an Ideal Has a Cyclic Basis

This paper focuses on the research of the existence of cyclic bases for the quotient ring modulo, a zero-dimensional ideal. Based on the necessary and sufficient condition for the existence of cyclic bases, it further deduces an equivalent condition for the existence of cyclic bases. In accordance with this condition, a new algorithm is proposed. The existence of a cyclic is determined by constructing a matrix and calculating its determinant in the algorithm. Compared with the original algorithm, it narrows the scope of the candidate element set and improves the computational efficiency of the algorithm.