 The free ring on one generator \(x\) must include \(x^2 , x^3\), etc. by multiplication, but these can be added and subtracted resulting in polynomials such as \(7x^3 -3x^2 +2x\) but without a constant term, with the exception of 0 itself. The distributivity law for rings means that a term such as \((7x+x^2 )(2x^3 +x)\) can be expanded as \(7x^2 +x^3 +14x^4 +2x^5\). It should now be clear that these are just ordinary polynomials with no constant term; in particular we are missing the zero-degree polynomial 1 and so this ring has no multiplicative identity. However it is a commutative ring even though we did not specify this. The free ring with identity on one generator introduces 1 as the multiplicative identity and becomes the ordinary one-variable polynomials since now we can form all the integers. Just as with monoids, the free ring with identity on two generators is not commutative, the polynomials \(xy\) and \(yx\) being distinct. The free commutative ring with identity on two generators however consists of the ordinary two-variable polynomials over the integers.