7. The number of distinct homomorphisms from the ring (Z, +,.) onto the ring (Ze, +,.) is (a) 0 (b) 1 (c) 2 (d) co (a) O (b) O (c) (d) O

Q: 2. Consider the following functions. Are these ring homomorphisms? If yes, prove it. If no, provide…

A: a) f:Z→Z given by f(x) = 3x For this function to be a ring homomorphism, it needs to satisfy two…

Q: a The mapping $ :C → M2 (R) given by (a + bi) = -b is a not one-to-one a ring homomorphism not a…

A: To find - The mapping ϕ : ℂ → M2ℝ given by ϕa + ib = ab-ba is ?     Definition used - A function is…

Q: 4) Let f : C – C be a homomorphism of rings with f(x) = x for all x E R. Show that f is either the…

A: If a function f is ring homomorphism from a ring R to another ring R'. Then We knew from the…

Q: Let (R, +,-) ring with properties X.XX, XER Cindempotent properties), show! @a = 0, x ER (b) x. y…

A:

Q: 4. Let R be a ring. (a) Let x, y, z € R. Prove that, if x | y and y | z, then x | z. (b) Prove that…

A: Let R be  a  ring .4) a)  Let  Suppose that   divides y and  y  divides z.We  have  to show that x…

Q: Consider the quotient ring R1 = Z3[a]/(x² + 1) and R2 = Z3[i] = {a + bi / a,b e Z3} with operations…

A: Consider the quotient ring R1=Z3x/x2+1  and  R2=Z3i=a+bi/a,b∈Z3 with operations…

Q: Suppose Φ is a ring homomorphism from Z ⨁ Z into Z ⨁ Z. Whatare the possibilities for Φ((1, 0))?

A: Let ϕ be a ring homomorphism from ℤ⊕ℤ into ℤ⊕ℤ. Then, we know that ϕ is a group homomorphism and a…

Q: Can you me please Give an example of a ring R and polynomials f (x) and g(x) in R[x] such that…

A:

Q: Let o : Z, → Zn be a nontrívial ring homomorphism. (a)What can you say about n ?( Explain).…

A: See the detailed solution below.

Q: Is the map y: C→ C defined by y(x + iy) = x² = y² a ring isomorphism of C? Is it a ring…

A: Given that , γ:ℂ→ℂ defined by , γx+iy=x2=y2

Q: Which of the following are ring (i): f: (2Z, +,-) → (Z, +,-), (ii): ƒ: (Z8, +,-) → (Z4, +,-), - 2x -…

A:

Q: 18. Let (R, ,) be a commutative ring with identity and let N denote the set of nilpotent clements of…

A: a). As given that N is nonempty as 0∈N.  an=0 (n= positive integer) Then r∈R ran=rnan=0 Let take a,…

Q: don't understand how phi(a+c)=a+c. Please explain. Thank you

A: Given

Q: M n of ring (5 Marks) (b) Let (R,+,) and (R', +', '') be any two rings, let f: RR' be a ring…

A:

Q: 14. Find all units of the ring C(R) = {ƒ : R → Rf is continuous}, where the operations are the usual…

A: A unit is an element which has multiplicative inverse. Using this definition we find the units in…