# getIdeals -- get component ideals from a monomial module

## Synopsis

• Usage:
getIdeals M
• Inputs:
• M, a monomial submodule of the ambient module over an exterior algebra
• Outputs:
• a list, list of ideals that determine the submodules whose direct sum is M

## Description

Let M be a submodule of F and let $\{g_1,g_2,\ldots,g_r\}$ be a graded basis of F. This method returns a list $L=\{I_1,I_2,\ldots,I_r\}$ such that $M=I_i g_i \oplus I_2 g_2 \oplus \cdots \oplus I_r g_r$.

Example:

 i1 : E = QQ[e_1..e_4, SkewCommutative => true] o1 = E o1 : PolynomialRing, 4 skew commutative variables i2 : m=matrix {{e_1*e_2,e_3*e_4,0,0,0},{0,0,e_1*e_2,e_2*e_3*e_4,0},{0,0,0,0,e_2*e_3*e_4}} o2 = | e_1e_2 e_3e_4 0 0 0 | | 0 0 e_1e_2 e_2e_3e_4 0 | | 0 0 0 0 e_2e_3e_4 | 3 5 o2 : Matrix E <--- E i3 : M=image m o3 = image | e_1e_2 e_3e_4 0 0 0 | | 0 0 e_1e_2 e_2e_3e_4 0 | | 0 0 0 0 e_2e_3e_4 | 3 o3 : E-module, submodule of E i4 : getIdeals M o4 = {ideal (e e , e e ), ideal (e e , e e e ), ideal(e e e )} 3 4 1 2 1 2 2 3 4 2 3 4 o4 : List