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Within the first 10 years of the demise of the Roman Empire, the Empire had managed to rule in the Mediterranean, Northern Africa, and Western Europe with a surprising degree of military success. A new century had begun under the emperor Trajan on January 1, 98; he would reign until his death on March 15, 117. The western half of the empire would experience difficulties for several more years. However, the East had not been totally defeated, and their control of the Mediterranean was solid.Q:

Dimension of a subspace of free modules

Let $R$ be a $k$-algebra, $I$ its ideal, and $M$ a finitely generated $R$-module. I am looking for the basis for the vector space $I\cdot M$. In my lectures, the lecturer claimed that there should be a basis of the form $IM^n$, where $n$ is a non-negative integer. But I couldn’t find it. Can anyone prove it?

A:

It’s easier to start with some elements not in $IM$, then take a spanning set in $IM$.
When we have $r_1,\ldots,r_n$ in $R$ with $r_1\cdots r_n=0$ we can write $r_1=a_1\cdots a_n$ with $a_i\in I$. Then we have $r_1m=\dots=r_nm=0$ and $m\in (I^{n+1})^\perp\subseteq I^nM$. It follows that $I^nM\subseteq I^{n+1}M^\perp$, and we are done.

When the Houston Texans take the field on Monday night, they’ll be playing a team looking to continue their dominance — the New England Patriots.

While the Patriots (4-1) are the early favorite to win Super Bowl LII, it’s clear that the Texans (3-2) are starting to make their mark in the AFC. Houston had a dominant comeback win over the Kansas City Chiefs in Week 4, and they made a statement Sunday against the Dallas Cowboys in a 36-28 win.

O.J. Anderson’s 13-yard touchdown run put the Texans up early, and their defense held strong. The Texans had another stellar

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