In algebra, a generic matrix ring of size n with variables
X
1
,
…
X
m
, denoted by
F
n
, is a sort of a universal matrix ring. It is universal in the sense that, given a commutative ring R and n-by-n matrices
A
1
,
…
,
A
m
over R, any mapping
X
i
↦
A
i
extends to the ring homomorphism (called evaluation)
F
n
→
M
n
(
R
)
.
Explicitly, given a field k, it is the subalgebra
F
n
of the matrix ring
M
n
(
k
[
(
X
l
)
i
j
|
1
≤
l
≤
m
,
1
≤
i
,
j
≤
n
]
)
generated by n-by-n matrices
X
1
,
…
,
X
m
, where
(
X
l
)
i
j
are matrix entries and commute by definition. For example, if m = 1, then
F
1
is a polynomial ring in one variable.
For example, a central polynomial is an element of the ring
F
n
that will map to a central element under an evaluation. (In fact, it is in the invariant ring
k
[
(
X
l
)
i
j
]
GL
n
(
k
)
since it is central and invariant.)
By definition,
F
n
is a quotient of the free ring
k
⟨
t
1
,
…
,
t
m
⟩
with
t
i
↦
X
i
by the ideal consisting of all p that vanish identically on any n-by-n matrices over k. The universal property means that any ring homomorphism from
k
⟨
t
1
,
…
,
t
m
⟩
to a matrix ring factors through
F
n
. This has a following geometric meaning. In algebraic geometry, the polynomial ring
k
[
t
,
…
,
t
m
]
is the coordinate ring of the affine space
k
m
and to give a point of
k
m
is to give a ring homomorphism (evaluation)
k
[
t
,
…
,
t
m
]
→
k
(either by the Hilbert nullstellensatz or by the scheme theory). The free ring
k
⟨
t
1
,
…
,
t
m
⟩
plays the role of the coordinate ring of the affine space in the noncommutative algebraic geometry (i.e., we don't demand free variables to commute) and thus a generic matrix ring of size n is the coordinate ring of a noncommutative affine variety whose points are the Spec's of matrix rings of size n (see below for a more concrete discussion.)
