Let $R = \mathbb{Z}$ and let $M = \mathbb{Z} / 4 \mathbb{Z}$ be an $R$-module. What is a primary decomposition for $M$?

Given random variables $\vec{x}, \vec{y} \in \mathbb{R}^n$, is it true that $I(\vec{x}: \vec{y}) \geq \sum_i I(x_i, y_i)$ My interpretation is that that collectively several variables should be able to predict another set of variables at least as good as individually. However, when I compute th...