up previous next
 2.11.1 Introduction to RINGELEM
An object of type RINGELEM in CoCoA represents an element of a ring.

To fix terminology about polynomials (elements of a polynomial ring): a polynomial is a sum of terms; each term is the product of a coefficient and power-product, a power-product being a product of powers of indeterminates. (In English it is standard to use monomial to mean a power-product, however, in other languages, such as Italian, monomial connotes a power-product multiplied by a scalar. In the interest of world peace, we will use the term power-product in those cases where confusion may arise.)

 Example
 ```/**/ Use R ::= QQ[x,y,z]; /**/ F := ***3xyz + xy^2***; /**/ F; xy^2 + 3xyz ------------------------------- Use R ::= QQ[x[1..5]]; Sum([x[N]^2 | N In 1..5]); x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2 + x[5]^2 ------------------------------- ```
CoCoA always keeps polynomials ordered with respect to the term-orderings of their corresponding rings.

The following algebraic operations on polynomials are supported:
```  F^N, +F, -F, F*G, F/G if G divides F, F+G, F-G,
```
where F, G are polynomials and N is an integer. The result may be a rational function.

 Example
 ``` Use R ::= QQ[x,y,z]; F := x^2+xy; G := x; F/G; x + y ------------------------------- F/(x+z); (x^2 + xy)/(x + z) ------------------------------- F^2; x^4 + 2x^3y + x^2y^2 ------------------------------- F^(-1); 1/(x^2 + xy) ------------------------------- ```