In this paper we study primality and primary decomposition of certain ideals which are generated by homogeneous degree 2 polynomials and occur naturally from determinantal conditions. Normality is derived from these results.2010 Mathematics Subject Classification. Primary 13C40, 13P10.

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In this paper we compute Gröbner bases for determinantal ideals of the form I 1 (XY ), where X and Y are both matrices whose entries are indeterminates over a field K. We use the Gröbner basis structure to determine Betti numbers for such ideals.2010 Mathematics Subject Classification. Primary 13P10; Secondary 13C40, 13D02.

In this paper we prove conditions for transversal intersection of monomial ideals and derive a simplicial characterization of this phenomenon. Theorem 3.2. As an application of this theorem we prove in Corollary 3.3 that for two ideals I and J in R intersecting transversally, the ideal I + J is resolved minimally by the complex M(I) ⊗ M(J) , if M(I) and M(J) denote minimal free resolutions of I and J respectively. Minimal free resolutions for ideals of the form I + J have an interesting structure when I and J intersect transversally and are supported simplicially; see 4.3. MONOMIAL IDEALSLet R = K[x 1 , x 2 , · · · , x n ], where x i 's are indeterminates over the field K. Let Mon(R) denote the set of all monomials in R. Every nonzero polynomial f ∈ R is a unique K-linear combination of monomials givenAn ideal I in R is said to be a monomial ideal if it is generated by monomials of R. We list down some standard facts on monomial ideals. 2010 Mathematics Subject Classification. 13C05; 13D02.

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