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Algebra-II: Companion notes

Abhay Soman

Section 9.6 Sums, products, and direct sums

Throughout this section we assume that \(M\) is a left module over a ring \(R\text{.}\) For some non-empty indexing set \(I\text{,}\) let \(M_i\) (\(i\in I\)) be left \(R\)-submodules of \(M\text{.}\)

Definition 9.6.1. (Sum of submodules).

The sum of \(R\)-submodules \(M_i\) is the submodule generated by \(\bigcup_{i\in I}M_i\text{.}\) It is denoted by \(\sum_{i\in I}M_i\text{.}\) Thus
\begin{equation*} \sum_{i\in I}M_i=\left\{\sum_im_i:m_i\in M_i\;\text{are such that all but finitely many}\;m_i\;\text{are zero}\right\} \end{equation*}

Definition 9.6.2. (Product of modules).

Let \((N_i)_{i\in I}\) be a family of left modules over a ring \(R\text{.}\) We give the Cartesian product \(\prod_{i\in I}N_i\) an \(R\)-module structure by defining addition and multiplication as follows.
\begin{equation*} (x_i)+(y_i)=(x_i+y_i)\quad\text{where}\; x_i,y_i\in N_i \end{equation*}
For any \(r\in R\) and any \((x_i)\in\prod_{i\in I}N_i\text{,}\) the scalar multiplication is defined by
\begin{equation*} r\cdot (x_i)=(r\cdot x_i) \end{equation*}

Definition 9.6.3. (Projection maps).

Let \((N_i)_{i\in I}\) be a family of left modules over a ring \(R\text{.}\) For each \(i\) we get the family of \(R\)-module homomorphism
\begin{equation*} \pi_i\colon\prod_{k\in I}N_k\rightarrow N_i\quad\text{given by}\;(x_k)_{k\in I}\mapsto x_i. \end{equation*}
An \(R\)-mod homomorphism \(\pi_i\) is called the projection homomorphism.

Definition 9.6.4. (Direct sum of modules).

Consider a family \((N_i)_{i\in I}\) of left modules over a ring \(R\text{.}\) The direct sum of \((N_i)_{i\in I}\) is the following submodule of \(\prod_{i\in I}N_i\text{.}\)
\begin{equation*} \bigoplus_{k\in I}N_k=\left\{(x_k)_{k\in I}\in\prod_{k\in I}N_k:\pi_i((x_k)_{k\in I})=0\;\text{except for a finite number of indices}\right\} \end{equation*}

Convention 9.6.5.

Consider \(R\) as a left module over itself. We denote by \(R^{(n)}\) an \(R\)-module of the direct sum of \(n\)-copies of \(R\text{,}\) i.e.,
\begin{equation*} R^{(n)}=\underbrace{R\oplus R\oplus\cdots\oplus R}_{n-\text{times}}. \end{equation*}

Definition 9.6.6. (Minimal generating subset and the rank of a module).

Let \(M\) be a finitely generated left \(R\)-module. Suppose that \(m_1,m_2,\ldots,m_r\in M\) are such that \(M=Rm_1+\cdots+Rm_r\text{.}\) We call \(\{m_1,\ldots,m_r\}\) a generating subset of \(M\text{.}\) By Well-ordering principle, there exists a minimal generating subset of \(M\text{,}\) such a generating subset is called a minimal generating subset of \(M\). The cardinality of a minimal generating subset of \(M\) is called the rank of \(M\). The rank of \(M\) is denoted by \({\rm rank}(M)\text{.}\)

Definition 9.6.7. (Free module of a finite rank).

An \(R\)-module \(M\) is said to be a free module of a finite rank if \(M\) is \(R\)-module isomorphic to \(R^{(n)}\) for some natural number \(n\text{.}\)

Definition 9.6.8. (\(R\)-linearly independence).

Let \(M\) be an \(R\)-mod, and let \(m_1,\ldots,m_k\in M\text{.}\) We say that \(m_1,\ldots,m_k\in M\) are \(R\)-linearly independent if \(r_1m_1+\cdots+r_km_k=0\) then \(r_1=r_2=\cdots=r_k=0\in R\text{.}\)
Elements \(x_1,\ldots,x_s\in M\) are said to be \(R\)-linearly dependent if \(x_1,\ldots,x_s\) are not \(R\)-linearly independent. In other words, \(x_1,\ldots,x_s\in M\) are \(R\)-linearly dependent if there are \(r_1,\ldots,r_s\in R\text{,}\) not all zero, such that \(r_1x_1+\cdots+r_sx_s=0\text{.}\)
Consider an abelian group \(\Z/n\Z\text{,}\) i.e., consider \(\Z/n\Z\) as a \(\Z\)-module. We show that any \([i]\in\Z/n\Z\) is \(\Z\)-linearly independent. Indeed, for any \([i]\in\Z/n\Z\) we have \(n\cdot[i]=[0]\text{.}\)
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