Examples

• Any field is an integral domain. So this includes Q,R, C, and Z_p, the rational numbers, the real numbers, the complex numbers, and the integers modulo a prime p.
• Any subring of a field is an integral domain. So this includes the ring of integers Z and the Gaussian integers Z[i].
• The ring k[x] of polynomials over another integral domain k form another integral domain (proved by looking at leading terms). So this gives lots more examples, like Z[x],C[x] or Q[x,y]
• The ring Z_m of integers modulo m for m> 3 composite is not an integral domain.
• The ring of 2x2 matrices with complex number coefficients is not an integral domain.