Polynomial ring $K[x]$

Let \(K\) be a commutative ring. The polynomial ring in the variable \(x\) over \(K\) is \[ K[x] \coloneqq \theset{\sum_{i=0}^n a_ix^i \st n\in \ZZ^{\geq 0},\ a_0,\ldots,a_n\in K}, \] where two polynomials are equal when their coefficients of \(x^i\) agree for every \(i\geq0\), with missing coefficients taken to be \(0\). Addition is coefficientwise, and multiplication is \(\bigl(\sum_i a_ix^i\bigr)\bigl(\sum_j b_jx^j\bigr)=\sum_k\bigl(\sum_{i+j=k}a_ib_j\bigr)x^k\).