We study the regularity of binomial edge ideals. For a closed graph G we show that the regularity of the binomial edge ideal JG coincides with the regularity of in lex (JG) and can be expressed in terms of the combinatorial data of G. In addition, we give positive answers to Matsuda‐Murai conjecture for some classes of graphs.

A method for computing the multigraded Hilbert depth of a module was presented in [16]. In this paper we improve the method and we introduce an effective algorithm for performing the computations. In a particular case, the algorithm may also be easily adapted for computing the Stanley depth of the module. We further present interesting examples which were found with the help of an experimental implementation of the algorithm [17]. Thus, we completely solve several open problems proposed by Herzog in [12].

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We show that the Stanley's Conjecture holds for an intersection of three monomial primary ideals of a polynomial algebra S over a field.