Let R be a finite ring and α ∈ R with α ≠ 0. If α is not a zero divisor, then α is a unit.
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Let R be a finite ring and α ∈ R with α ≠ 0. If α is not a zero divisor, then α is a unit.
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- Prove that if a is a unit in a ring R with unity, then a is not a zero divisor.Let I be an ideal in a ring R with unity. Prove that if I contains an element a that has a multiplicative inverse, then I=R.Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4
- 21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.27. If is a commutative ring with unity, prove that any maximal ideal of is also a prime ideal.