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

====Comments====
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 $).

The term  "scalar"  comes from the original meaning as a quantity which can be completely specified by one (real) number.

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|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 $.

A scalar operator on, say, a complex Banach space is a scalar multiple of the identity operator.

Given a left module  $  M $
over a ring  $  R $
and an  $  R $-
algebra  $  S $, 
one forms the [[Tensor product|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.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,   "Algebra" , '''1''' , Wiley  (1982)  pp. 70</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Rektorys (ed.) , ''Applicable mathematics'' , Iliffe  (1969)  pp. 270; 290</TD></TR></table>