# ApCoCoA-1:NC.IsFinite

From ApCoCoAWiki

## 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.

### See also