A Polarization Operation for Pseudomonomial Ideals
نویسنده
چکیده
Pseudomonomials and ideals generated by pseudomonomials (pseudomonomial ideals) are a central object of study in the theory of neural rings and neural codes. In the setting of a polynomial ring, we define the polarization operation ρ sending pseudomonomials to squarefree monomials and a further polarization operation P sending pseudomonomial ideals to squarefree monomial ideals. We show for a pseudomonomial ideal I, in a polynomial ring R, that • A pseudomonomial f is in I if and only if ρ(f) is in P(I). • P(I) is generated by the polarizations of the minimal pseudomonomials in I. • The prime ideals in the unique minimal primary decomposition of P(I) are the polarizations P(p) of the prime ideals p in the unique minimal primary decomposition of I. Furthermore, I is Cohen-Macaulay if and only if P(I) is.
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