CHAPTER 2
NC functions and their difference-differential
calculus
2.1. Definition of nc functions
Let R be a commutative ring with identity. For M a module over R, we define
the nc space over M
Mnc =

n=1
Mn×n.
For X
Mn×n
and Y
Mm×m
we define their direct sum
X Y =
X 0
0 Y

M(n+m)×(n+m).
Notice that matrices over R act from the right and from the left on matrices over
M by the standard rules of matrix multiplication and by the action of R on M: if
X
Mp×q
and T
Rr×p,
S
Rq×s,
then
TX
Mr×q,
XS
Mp×s.
A subset Ω Mnc is called a nc set if it is closed under direct sums; explicitly,
denoting Ωn = Ω Mn×n, we have X Y Ωn+m for all X Ωn, Y Ωm. We
will sometimes use the convention that Mn×0, M0×n, and Ω0 = M0×0 consist
each of a single (zero) element, the empty matrix “of appropriate size”, and that
X Y = Y X = X Ωn for n N, X Ωn, and Y Ω0.
In the case of M =
Rd,
we identify matrices over M with d-tuples of matrices
over R:
(
Rd
)p×q

=
(
Rp×q
)d
.
Under this identification, for d-tuples X = (X1,...,Xd)
(Rn×n)d
and Y =
(Y1,...,Yd)
(Rm×m)d,
X Y =
X1 0
0 Y1
, . . . ,
Xd 0
0 Yd

R(n+m)×(n+m)
d
;
and for a d-tuple X = (X1,...,Xd)
(Rp×q)d
and matrices T
Rr×p,
S
Rq×s,
TX = (TX1,...,TXd)
(
Rr×q
)d
, XS = (X1S,...,XdS)
(
Rp×s
)d
.
Let M and N be modules over R, and let Ω Mnc be a nc set. A mapping
f : Ω Nnc, with f(Ωn) N
n×n,
n = 0, 1,..., is called a nc function if f satisfies
the following two conditions:
f respects direct sums:
(2.1) f(X Y ) = f(X) f(Y )
for all X, Y Ω.
15
http://dx.doi.org/10.1090/surv/199/02
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