Transversal intersection and sum of polynomial ideals

In this paper we derive some conditions for transversal intersection of polynomial ideals and present some examples. Finally, as an application of the results proved, we compute the Betti numbers of ideals of the form I1(XY) +J, where X and Y are some generic or generic symmetric matrices and J is the ideal generated by the 2×2 minors of the matrix consisting of any two rows of X.