Ideal of a ring (reviewed)

If $R$ is a ring, a subset $I\subseteq R$ is an ideal of $R$ if $I$ is a subgroup of $R$ for $+$ and for all $a\in I$ and all $r\in R$, $$ r\cdot a\in I \qquad\text{and}\qquad a\cdot r\in I.$$

In a polynomial ring $R[X_1,\dots,X_n]$, an ideal is homogeneous if it can be generated by homogeneous polynomials.