Remark

(1) Let $E$ be an $A$-module, $(F_i)_{i \in I}$ a family of $A$-modules and for all $i \in I$ let $u_i : E \to F_i$ be a linear mapping. Every system of linear equations

where the $y_i \in F_i$ are given, is equivalent to a single linear equation $u(x) = y$, where $u$ is the mapping $x \mapsto (u_i(x))$ of $E$ into $F = \prod_{i \in I} F_i$ and $y = (y_i)$. The system (29) is called homogeneous if $y_i = 0$ for all $i \in I$.

(2) Suppose that $E$ admits a basis $(a_\lambda)_{\lambda \in L}$; if we set $u(a_\lambda) = b_\lambda$ for all $\lambda \in L$, to say that $x = \sum_{\lambda \in L} \xi_\lambda a_\lambda$ satisfies the equation $u(x) = y_0$ is equivalent to saying that the family (of finite support) $(\xi_\lambda)_{\lambda \in L}$ of elements of $A$ satisfies the relation

Conversely, looking for families $(\xi_\lambda)_{\lambda \in L}$ of elements of $A$ of finite support satisfying (30), is equivalent to solving the linear equation $u(x) = y_0$, where $u$ is the unique linear mapping of $E$ into $F$ such that $u(a_\lambda) = b_\lambda$ for all $\lambda \in L$ (§ 1, no. 11, Corollary 3 to Proposition 17).

(3) A linear equation $u(x) = y_0$ is called scalar when $F = A_s$ and therefore $u$ is a linear form on $E$ and $y_0$ a scalar. If $E$ admits a basis $(a_\lambda)_{\lambda \in L}$, it follows from Remark (2) that such an equation may also be written as

where the family of scalars $(\alpha_\lambda)$ is arbitrary and where it is understood that the family $(\xi_\lambda)$ must have finite support. In general, by the solution (in $A$) of a system of scalar linear equations where $\alpha_{\lambda_i} \in A$ and $\eta_i \in A$, is understood a family $(\xi_\lambda)_{\lambda \in L}$ of elements of $A$ of finite support and satisfying (32); the $\alpha_{\lambda_i}$ are called the coefficients of the system of equations and the $\eta_i$ the right hand sides. The solution of such a system is equivalent to that of the equation $u(x) = y$, where $y = (\eta_i)$ and $u : A_s^{(L)} \to A_s^I$ is the linear mapping

(4) A linear system (32) is also called a system of left scalar linear equations when it is necessary to avoid confusion. A system of equations

is likewise called a system of right scalar linear equations; such a system can immediately be transformed into a system (32) by considering the $\xi_{\lambda}, \eta_i$ and $\alpha_{\lambda_i}$ as belonging to the opposite ring $A^0$ to A.