Let $K$ be a field such that there exists an algorithm for factoring a polynomial over $K$ into the product of irreducible polynomials. For example, the field of rational numbers $\mathbb{Q}$ is such a field. Let $K[[x_1,\cdots,x_n]]$ be the formal power series ring over $K$. Let $I$ be an ideal of $K[[x_1,\cdots,x_n]]$ generated by $f_1,\cdots,f_m$. According to the well-known theorem due to E. Noether, there exists a primary decompositon of $I$. Namely $I = Q_1\cap\cdots Q_r$, where each $Q_i$ is a primary ideal and this decomposition is irredundant. Is there an algorithm to find generators of each $Q_i$? If yes, I would like to know the reference. Thanks in advance.
Motivation There exists such an algorithm if $I$ is an ideal of a polynomial ring over $K$. So it's natural to ask the above question.