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