Assume that $R$ is  a unital ring or  a  complex or real   (Banach or  $C^{*}$) algebra.

We  define  a relation $M$  on $R$ as follows: $$a\;M  b \;\;\;  \text{iff}\;\; a=xy,\;b=yx \;\; \text{for  some}\;\; x,y\in R$$  It is  a reflexive and symmetric (but  not transitive )  relation. We  define an equivalent relation $\simeq$ on $R$ as  follows:

$a\simeq b$ if there are $p_{i}\in R,\;i=0,1,\ldots,n$  with $$\begin{cases}a=p_{0},\;\; b=p_{n},&\\ p_{i} \; M\; p_{i+1}\end{cases}$$

The space of **nilpotent** elements of $R$, denoted by $N(R)$, is  a saturated subset of $R$ while the space of idempotent elements is not necessarily a saturated subset.

Notation: $M_{n}(R)$ is the space of $n\times n$  matrices with entries in $R$.

The   natural  mapping $M_{n}(R) \to  M_{n+1} (R)$ with $A \mapsto A\oplus 0$ sends  nilpotent elements to   nilpotent elements.  Moreover the above equivalent relation is  preserved under this map. 

We  consider $\bigcup_{n=1}^{\infty} N(M_{n}(R))$, the union of  all  nilpotent  matrices of all size. The equivalent relation  $\simeq$ has a natural extension to the later space: $A\simeq  B$ if there are natural numbers  $k,p$  such that $A\oplus 0_{k} \simeq B\oplus  0_{p}$. The later are  the zero matrices of  size $k,p$, respectively. 

This enable us to equip $\bigcup_{n=1}^{\infty} N(M_{n}(R))/\simeq$  to  an Abelian semi group structure, via the usual matrix-sum.(Note that for two  nilpotent  elements  $a,b$  with $ab=ba=0$,  $a+b$ is  again  a  nilpotent element. On the other hand every two elements of the above  quotient space have two representation $A,B$  with $AB=BA=0$). Because,   for every $a\in R$  we  have $\begin{pmatrix} a&0\\0&0  \end{pmatrix} \simeq \begin{pmatrix} 0&0\\0&a \end{pmatrix}$. 

> The Grothendick group of this  semi group  is  denoted  by $NK(R)$.

**Questions:**
> What is  an example of  a $C^{*}$ algebra  $A$ for which $NK(A)$  is a  non trivial group? Is there  a  commutative  $C^{*}$  algebra $A$ with  nontrivial $NK(A)$.

**Note 1** The mapping $A\mapsto  NK(A)$ is  realy  a functor on the category of  rings or  algebra. according to Gelfand  Naimark duality this could  be  counted  as  a  functor on the category of  compact  haussdorff topological  space




**Note 2:** This  post  is  inspired by the construction in algebraic K theory and the following two posts

http://mathoverflow.net/questions/231328/the-saturation-of-murray-von-neumann-relation


http://math.stackexchange.com/questions/1661660/the-reduction-of-nilpotency-order-of-nilpotent-elements-of-c-algebras