in the following rings find all units, Zero divisors and idompotent elements and determine whether the ring is a field.
a)Z2
b)Z5
c)Z10
NB z is integers( positive or negatives including zero
z is integers( positive or negatives including zero
To find the units, zero divisors, and idempotent elements in a ring, we can apply the definitions of each term.
1. Units: In a ring, units are elements that have multiplicative inverses. To find the units in the given rings, we need to check which elements have inverses.
a) Z2: The elements of the ring Z2 are {0, 1}. The only element with a multiplicative inverse in this ring is 1, which is its own inverse. Therefore, the unit in Z2 is 1.
b) Z5: The elements of the ring Z5 are {0, 1, 2, 3, 4}. To find the units, we check which elements have inverses modulo 5. Using modular arithmetic, we can see that the units in Z5 are {1, 2, 3, 4}. Each of these elements has a unique multiplicative inverse in the ring.
c) Z10: The elements of the ring Z10 are {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. To find the units, we check which elements have inverses modulo 10. Using modular arithmetic, we can see that the units in Z10 are {1, 3, 7, 9}.
2. Zero Divisors: In a ring, zero divisors are non-zero elements that multiply to give zero. To find the zero divisors in the given rings, we can check which elements, when multiplied by another non-zero element, result in zero.
a) Z2: There are no zero divisors in Z2 since the only non-zero element is the unit 1, and multiplying it by any non-zero element gives 1.
b) Z5: Z5 does not have zero divisors either since multiplying any non-zero element by another non-zero element in Z5 does not give zero.
c) Z10: The zero divisors in Z10 are {0, 2, 4, 5, 6, 8}. These elements, when multiplied by a non-zero element in Z10, result in zero.
3. Idempotent Elements: In a ring, idempotent elements are elements that satisfy x^2 = x. To find the idempotent elements in the given rings, we need to check which elements, when squared, give the same element.
a) Z2: The idempotent elements in Z2 are {0, 1}, since both 0^2 = 0 and 1^2 = 1.
b) Z5: In Z5, there are no idempotent elements because for any non-zero element x, x^2 is not equal to x.
c) Z10: The idempotent elements in Z10 are {0, 1, 5, 6}. These elements satisfy x^2 = x.
Determining whether the given rings are fields:
A field is a ring in which every non-zero element is a unit. From our analysis above, we can see that:
a) Z2 is not a field, as 0 is not a unit.
b) Z5 is a field, as every non-zero element is a unit.
c) Z10 is not a field, as not every non-zero element is a unit.