. Let R be the ring of all complex numbers and R' the ring of all 2 x2 matrices of the form a b b a where a and b are real | numbers. Then the mapping f : R→R' defined as а b f (a + ib) = - b is a homomorphism. a

Q: 2) The algebraic closure C of the complex number field C is C itself. O True. C is algebrdi cally…

A: The question is solved in step 2.

Q: Q1. a) Give an example of a semi-ring that is not a semi -algebra. b) Give an example of a…

A:

Q: 1) The r ing of in tegers (Z) is not ideal of Yeal umber (R) n O True Fulse

A:

Q: Show that the correspondence x --> 5x from Z5 to Z10 does not preserveaddition.

A: Given:Z5 and Z10. We have to show that the correspondence x→5x from Z5 to Z10 does not preserve…

Q: prove The intersection of two maximal ideals is maximal.

A:

Q: Give an example of a ring that has exactly two maximal ideals

A: To Do: Give an example of a ring that has exactly two maximal ideals.

Q: Show that a homomorphism of rings whose domain is a field is either injective or the zero…

A:

Q: Determine all ring homomorphism from Zn to Zn

A: (if n is not prime) No. of ring homomorphism from Zn to Zn = No. of Idempotent elements in Zn.

Q: Let R be a finite commutative ring with identity. Then every prime ideal of R is maximal True O…

A: To prove that every prime ideal of R is maximal.

Q: in a commutative ring with unit element, prove that the relation " a is an associate of b " is an…

A:

Q: Prove that gcd(a, b) is the least natural number representable in the form ax + by with x, y ∈ Z.

A: We have to show that gcd(a, b) is the least natural number representable in the form ax + by with x,…

Q: 4. Prove that a zero divisor in a ring cannot be a unit.

A:

Q: A ring, R, in which r^2 = r for all elements r in R is called a Boolean ring. In any Boolean ring,…

A: A ring is said to be commutative if r.s = s.r for all r,s in R

Q: Find all ring homomorphisms from Z_6 to Z_6

A:

Q: Let R = %3 { la, b e z}and let p:R - Zbe defined by : 0(1 ) = - . 1) is a ring a) Homomorphism. b)…

A: The solution is given by using definitions of homomorphism, isomorphism and kernel as follows

Q: 10. By using the Fundamental Homomorphism Theorem, prove the following statements: (a) (Z x Z)/((4,…

A: We will solve all the four parts for you, using basic knowledge of group isomorphisms.

Q: 5) Let (R,+) and (R.+,) be any two rings, let f: R R be a ring homomorphism Ker () (0), then a) f is…

A:

Q: 7. Aring with identity (R. +..) is commutative if: (a) Ris an integral domain () (a+ b) - a + Zab +…

A:

Q: Show that every ring can be embedded in a ring with identity.

A: We have to show that every ring can be embedded in a ring with identity.

Q: Give an example of a commutative ring without zero-divisors that is not an integral domain.

A: Integral domain: Let D be a ring. Then D is an integral domain, provided these conditions hold: 1. D…

Q: Let x, y, z be elements of the real numbers. Use the field axioms and ordered axioms to prove -0=0.

A: We make use of the following field axioms:

Q: Prove that F2  is a field

A: Given: F2 To Prove: F2 is a field.

Q: What is a ring homomorphism in simple terms?

A:

Q: Prove or disprove that characteristics of simple ring is either zero or prime

A: To Prove or disprove that characteristics of simple ring is either zero or prime.

Q: tegral domain with more t an integral domain is an Eve ring is a subring of so

A:

Q: Let G= Find all the conjugate classes of (G,.). (i.e G={1,a,b,a.b}).

A:

Q: Q1: Prove that every finite integral domain is field?

A: Let D is a finite integral domain. we will show D is a field. Let a∈D,a≠0. If a=1,a-1=1 and a is a…

Q: is union of two ideal rings R, an idea of R? prove or give counter example

A:

Q: Prove that R is a complete ordered field.

A: Any set F which satisfies the following 8 axioms is said to be a complete ordered field. If…

Q: Find a bijection from the set of positive integers Z+ to the set of integers Z (Thus, conclude Z+ =…

A: the set of positive integers Z+ to the set of integers Z . the function is defined..

Q: Let R be a ring with no zero divisors, such that x^3 = x for each x in R(Real Number). Prove that…

A: Let R be a ring with no zero divisors  such that  We  have , We  have to  prove that  Characteristic…