# Scalar

A quantity all values of which can be expressed by one (real) number. More generally, a scalar is an element of some field.

Generally, the setting is that of a (ground) field $F$( more generally, a ring $R$) and a vector space $V$( of functions, vectors, matrices, tensors, etc.) over it (more generally, a module $M$). The elements of $F$( respectively, $R$) are called scalars. If $V$( respectively, $M$) is an algebra with unit element $e$, the elements $\lambda e$, $\lambda$ in $F$( respectively, $R$) are also called scalars. For example, one sometimes speaks of the ( $n \times n$)- matrices $\mathop{\rm diag} ( \lambda \dots \lambda )$ as scalar matrices. The scalar multiples of an element $v \in V$( respectively, $m \in M$) are the elements $\lambda v$, $\lambda \in F$( respectively, $\lambda m$, $\lambda \in R$).
A scalar field on a manifold $M$ is a function on $M$; that is, a scalar field, or field of scalars, is a tensor field (cf. Tensor bundle) of rank $( 0, 0)$. These are the scalars in the algebra of tensor fields on $M$ over the ring of functions on $M$.
Given a left module $M$ over a ring $R$ and an $R$- algebra $S$, one forms the tensor product $S \otimes _ {R} M$. This is a module over $S$. The module $S \otimes _ {R} M$ is said to be obtained from $M$ by extension of scalars.