Subset of Module Containing Identity is Linearly Dependent

Theorem

Let $G$ be a group whose identity is $e$.

Let $\left({R, +, \circ}\right)$ be a ring whose zero is $0_R$.

Let $\left({G, +_G, \circ}\right)_R$ be an $R$-module.

Let $H \subseteq G$ such that $e \in H$.

Then $H$ is a linearly dependent set.

Proof

From Scalar Product with Identity, $\forall \lambda: \lambda \circ e = e$.

Let $H \subseteq G$ such that $e \in H$.

Consider any sequence $\left \langle {a_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ in $H$ which includes $e$.

So, let $a_j = e$ for some $j \in \left[{1 \,.\,.\, n}\right]$.

Let $c \in R \ne 0_R$.

Consider the sequence $\left \langle {\lambda_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ of elements of $R$ defined as:

$\lambda_k = \begin{cases} c & : k \ne j \\ 0_R & : k= j \end{cases}$

Then:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \sum_{k \mathop = 1}^n \lambda_k \circ a_k\)	\(=\)	\(\displaystyle \lambda_1 \circ a_1 + \lambda_2 \circ a_2 + \cdots + \lambda_j \circ a_j + \cdots + \lambda_n \circ a_n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle 0_R \circ a_1 + 0_R \circ a_2 + \cdots + c \circ e + \cdots + 0_R \circ a_n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle e + e + \cdots + e + \cdots + e\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle e\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

Thus there exists a sequence $\left \langle {\lambda_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ in which not all $\lambda_k = 0_R$ such that $\displaystyle \sum_{k \mathop = 1}^n \lambda_k \circ a_k = e$,.

Hence the result.

$\blacksquare$

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 27$
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 7.33$