We find an explicit expression of the associated primes of monomial ideals as a colon by an element [Formula: see text], using the unique irredundant irreducible decomposition whose irreducible components are monomial ideals. An algorithm to compute [Formula: see text] is given using Macaulay2. For squarefree monomial ideals the problem is related to the combinatorics of the underlying clutter or graph. For ideals of Borel type, the monomial [Formula: see text] takes a simpler form, and we classify when [Formula: see text] is unique.
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Ambhore et al. (2024) studied this question.