Definition 27: The characteristic of a ring R with 1 is the smallest nonzero number n such that 0=1+1+.,.+1(n times) if such an n exists: otherwise the characteristic is defined to be 0. We denoted by char(r). Theorem 6.32: Let p be the characteristic of a a ring r with 1(e). Then following results hold. ()For VaER, pa=0. And if r is an integral domain, then p is the smallest nonzero number such that 0=la. where a=0. (2)IfR is an integral domain, then the characteristic is either 0 or a prime number.•Definition 27: The characteristic of a ring R with 1 is the smallest nonzero number n such that 0 =1 + 1 + ···+ 1 (n times) if such an n exists; otherwise the characteristic is defined to be 0. We denoted by char(R). Theorem 6.32: Let p be the characteristic of a ring R with 1(e). Then following results hold. (1)For aR, pa=0. And if R is an integral domain, then p is the smallest nonzero number such that 0=la, where a0. (2)If R is an integral domain, then the characteristic is either 0 or a prime number