Let I be a monomial ideal of a polynomial ring \(R=K[x_1,\ldots ,x_n]\) over a field K, and let \(\textrm{sgn}(I)\) be its signature ideal. If I is not a principal ideal, we show that the depth of R/I is the depth of \(R/\textrm{sgn}(I)\) , and the regularity of \(R/\textrm{sgn}(I)\) is at most the regularity of R/I. For ideals of height at least 2, we show that the associated primes of I and \(\textrm{sgn}(I)\) are the same, and we show that I is Cohen–Macaulay (resp. Gorenstein) if and only if \(\textrm{sgn}(I)\) is Cohen–Macaulay (resp. Gorenstein), and furthermore, we show that the v-number of \(\textrm{sgn}(I)\) is at most the v-number of I and compare the irreducible decompositions of I and \(\textrm{sgn}(I)\) . We give an algorithm to compute the signature of a monomial ideal using Macaulay2, and an algorithm to examine given families of monomial ideals by computing their signature ideals and determining which of these are Cohen–Macaulay or Gorenstein.
