- Let R be a commutative ring with unity of prime characteristic p. Show that the map op : R →R given hu Op(a) = aP is a homomorphism (the Frobenius homomorphism).
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
Number 20
![ms, then the
are rings, and if o :
composite function o : R -→ R" is a homomorphism. (Use Exercise 49 of Section 13.)
20. Let R be a commutative ring with unity of prime characteristic p. Show that the map Op : R → R given hy
Op(a) = aP is a homomorphism (the Frobenius homomorphism).
21. Let R and R' be rings and let o : R → R' be a ring homomorphism such that o[R] 7 {0}. Show that if R has
unity 1 and R has no 0 divisors, then (1) is unity for R'.
22. Let ø : R → R' he a ring ho
hom](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb4dd8e23-ab66-4b24-8e54-a64daec9031c%2F493af593-9ffd-4f9e-96fc-dd2dee2adf11%2Fut7k7lz_processed.jpeg&w=3840&q=75)
Homomorphic ring means F:R---->R' and a, b belongs to R then
f(a+b) = f(a)+(b) and
f(ab) = f(a) * f(b)
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