Prove that R contains a zero divisor if and only if R' does.
To prove that R contains a zero divisor if and only if R' does, we first need to establish the definitions of zero divisor and R'.
Zero Divisor: In a ring R, an element a is called a zero divisor if there exists a non-zero element b in R such that ab = 0 or ba = 0.
R': Given a commutative ring R, R' denotes the set of nonzero elements of R along with the zero element.
Now, let's prove the statement:
1. If R contains a zero divisor, then R' does:
If a is a zero divisor in R, then there exists a non-zero element b in R such that ab = 0 or ba = 0. Since a is not equal to 0, it must be an element of R'. Likewise, b is also an element of R' because it is non-zero. Thus, R' contains a zero divisor.
2. If R' contains a zero divisor, then R does:
Suppose R' contains a zero divisor, let's say c. Since c is a zero divisor in R', there exists a non-zero element d in R such that cd = 0 or dc = 0. Since c is non-zero, it is an element of R. Similarly, d is also an element of R' since it is non-zero. Thus, R contains a zero divisor.
By proving both implications, we have shown that R contains a zero divisor if and only if R' does.