# ApCoCoA-1:NC.IsFinite

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## NC.IsFinite

Check whether a K-algebra R/<M> has finite K-dimension.

### Syntax

```NC.IsFinite(M:LIST):BOOL
```

### Description

Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.

Please set non-commutative polynomial ring (via the command Use) before calling this function. For more information, please check the relevant commands and functions.

• @param M: a LIST of words (or terms) in the defining ring. Note that each word is represented as a LIST, and that each element in the LIST involves only one indeterminate or none (a constant). For instance, the word x[2]y[1]x[2]^2 is represented as the LIST [x[2], y[1], x[2]^2].

• @return: a BOOL value which is True if K-vector space R/<M> is finite and False otherwise.

#### Example

```Use ZZ/(2)[x[1..3]];
M:=[[x[3]^3], [x[1], x[2]], [x[2]^2], [x[1]^2], [x[2], x[3], x[1]],
[x[1], x[3], x[1]], [x[1], x[3]^2], [x[2], x[3], x[2], x[3]], [x[1],
x[3], x[2], x[3]]];
NC.IsFinite(M);

False
-------------------------------
The output at the ApCoCoAServer is as follows.
Found 2 cycles:
[3,2,1]  [2,1,3]  [1,3,2]
[3,3,2]  [3,2,3]  [2,3,3]
The group has polynomial growth of at most degree 2.

```