# module quotients, saturation, and annihilator

## Module Quotients

There are two notions associated to the colon construction for modules.

Similar to the case of ideals, the quotient of two $R$-modules $M, N$ contained in the same ambient module is an ideal $M:N$ of elements $f\in R$ such that $f N \subset M$. This is equivalent to the annihilator of the quotient module $(M+N)/M$.

The quotient of an $R$-module $M\subset F$ with respect to an ideal $J\subset R$ is the module $M:_F J$ of elements $f\in F$ such that $J f\subset M$.

## Saturation of Modules

The saturation of an $R$-module $M\subset F$ with respect to an ideal $J\subset R$ is an $R$-module $M:_F J^\infty$ of elements $f\in F$ such that $J^N f\subset M$ for some $N$ large enough. If the ideal $J$ is not given, the ideal generated by the variables of the ring $R$ is used.

If $M=M:_F J^\infty$ (or, equivalently, $M=M:_F J$), we say that $M$ is saturated with respect to $J$. We can use this command to remove graded submodules of finite length.

 i1 : R = ZZ/32003[a..d]; i2 : m = ideal vars R o2 = ideal (a, b, c, d) o2 : Ideal of R i3 : M = R^1 / (a * m^2) o3 = cokernel | a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 | 1 o3 : R-module, quotient of R i4 : M / saturate 0_M o4 = cokernel | a a3 a2b a2c a2d ab2 abc abd ac2 acd ad2 | 1 o4 : R-module, quotient of R

## Module Annihilators

The annihilator of an $R$-module $M$ is the ideal $\mathrm{ann}(M) = \{ f \in R | f M = 0 \}$.

 i5 : R = QQ[a..i]; i6 : M = cokernel genericMatrix(R,a,3,3) o6 = cokernel | a d g | | b e h | | c f i | 3 o6 : R-module, quotient of R i7 : annihilator M o7 = ideal(c*e*g - b*f*g - c*d*h + a*f*h + b*d*i - a*e*i) o7 : Ideal of R

You may also use the abbreviation ann:

 i8 : ann (M/(a*M)) o8 = ideal (a, c*e*g - b*f*g - c*d*h + b*d*i) o8 : Ideal of R