Let R, S be rings with unity and o : R→ S a ring homomorphism. Show that if o(1R) is a unit, then (1R) = ls.

Q: Let R be a ring with unity 1 and char (R) = 4. %3D Then R contains a subring isomorphic to Q ZO Z3 O

A: IN the given question,  Given that: R is a ring with unity 1 and char(R)=4. we have to find: we have…

Q: Let I be an ideal in a ring R with unity. Show that I = R if and only if I contains a unit.

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Q: Let R be a ring with unity. Show that (a) = { £ xay : x, y e R }. finite

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Q: Let S be a ring. Determine whether S is commutative if it has the following property: whenever æy =…

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Q: The ring Zs(i) has no proper ideals True False

A: Given statement is false. Justification is given in step two.

Q: Let R be a ring with unity 1. Show that S = {n· 1 | nE Z} is a sub- ring of R.

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Q: The ring Z is isomorphic to the ring 3Z

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Q: Let A e M2(Q). 2 Define a ring homomorphism ф: Q] —> М2(Q), f(x) → f(A). (i) Show that A² – 2A – 3 =…

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Q: 4. Let y: R→ S be a ring homomorphism. Prove that ': R[X] → S[X] given by soʻ (ao + a1X + ..a,X") =…

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Q: a 2b Let Z[√√√2] = {a+b√2 \a, beZ} and let H = { [ b Show that Z[√2] and Hare isomorphic as rings. a…

A: We will be solving Q2 as mentioned. Given that ℤ2=a+b2:a,b∈ℤ and H=a2bba: a,b∈ℤ. Let, R and R' be…

Q: 1. Let R be a ring with the additive identity 0. Prove that for any a E R, 0- a = 0.

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Q: Let T be ring containing elements e, f, both # 0T, such that e + f = 1r , e² = e, f² = f , and e · f…

A: We are given e+f = 1T,e2=e,f2=f and e.f=0T where T is the rings containing elements e,f both not…

Q: Let R be a ring with unity 1R and let S be a subring of R containing 1R. If r∈R is a unit of R and…

A: Let R be a ring with unity 1R and let S be a subring of R containing 1R. If r∈R is a unit of R and…

Q: Let 1={0,2} CZ. Prove if ring Z,/l is isomorphic with a ring Z,

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Q: 3. Explain why the polynomial rings R[r] and C[r] are not isomorphic.

A: This is a problem of Abstract Algebra.

Q: Let R=Z and R'= set of all even integers. Then %3D (R', +, *) is a ring, where a* b= ab V a, be R'.…

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Q: If is a homomorphism from the ring R to the ring R' , show that; a) (0)=0 b) (−r)= −(r)for all…

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Q: Let R be a ring with unity. Show that (a) = { £ xay: x, y e R }. finite

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Q: Let R be a ring with unity. Show that (a) = { E xay : x, y e R }. finite

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Q: Let R be a commutative ring with unity and let N= {a e R| a" = 0 for some n e Z"; n> 1} Show that N…

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Q: Let R be a ring with 1. Show that R[z]/ (x) ~ R.

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Q: If u is finitely additive on a ring R; E, F eR show µ(E) +µ(F) = µ(EU F)+µ(EnF) %3D

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Q: The ring Z3[i] has no proper ideals O True O False

A: Yes it is true .

Q: Let S be a ring. Determine whether S is commutative if it has the following property: whenever ry =…

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Q: Let R be a ring with m elements. Show that the characteristic of Rdivides m.

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Q: If R is a noncommutative ring with unity and x, y ∈ R, compute the product x(x + y)(x − y)y.

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Q: The ring Zs[i] has no proper ideals

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Q: 2. Let R be a ring: The center of R is the set 3XER: ax= xa vae R? Prove that the center of a ring…

A: Let R be a ring .We have to show that centre of ring is a subring of R

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: 7. The number of distinct homomorphisms from the ring (Z, +,.) onto the ring (Ze, +,.) is (a) 0 (b)…

A: Number of distinct onto homomorphism is

Q: Let R be a commutative unitary ring and let M be an R-module. For every r ERlet rM = {rx; x E M} and…

A: We shall solve first question only as you have asked more than one different question as per company…

Q: 15. Let R be a ring, I, J be ideals of R and f: RR/I x R/J be the function defined by f(a) = (a +…

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Q: a Let S= { : a,b e R}. Show that : C→ S defined by %3D a b is a ring homomorphism. a $(a + bi) = -b…

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Q: Let R be a ring with unity and assume a ∈ R is a unit. Prove that a is not nilpotent.

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Q: Let R be a ring with unity 1, and S = {n.1 : n E Z} . Then S'is Ra subring of Rnot a subring of

A: Let, x ,y in S. So, x = n•1 and, y = m•1 for some n, m in Z.

Q: Let R be a ring with unity 1, and S = {n.1 : n E Z}.Then S'is Ra subring of Rnot a subring of

A: A non-empty set R with two binary operations addition(+) and multiplication(·) is called ring if it…

Q: 3. Let R be a ring and b E R be a fixed element. Let and prove that T is a subring of R T = {rb | r…

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Q: Q2) Let(M₂ (R), +..) be a ring. Prove H = {(a) la, b, c = R}is a subring of (M₂ (R), +,.).

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Q: R→s be a ring homomorphism. ve that if K is a subring of then R f(K) is a subring of S ve that c is…

A: Given, if f:R→S be a ring homomorphism. (i) To prove that if K is a subring of R then f(K) is a…

Q: Is the idcal (x² + 1, x + 3) C Z[x] a principal idcal? Explain. The ring Z[x]/(x² +1, x+3) is…

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Q: Find all the principal ideals of the ring (Z6, O, 0 0 ).

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Q: Iff is a ring homomorphism from Zm to Zn such that f (1)=b, then b*+2 = b*. False True

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Q: The map f: Z→ Z,o given by f(x)= 2x is a ring homomorphism. Select one. True False

A: SINCE YOU HAVE ASKED MULTIPLE QUESTIONS IN SINGLE REQUEST, WE WILL BE ANSWERING ONLY THE FIRST…

Q: a Let S = { : a,b e R } . Show that Ø : C→ S defined by a b is a ring homomorphism. a Ф(а + bi)- a

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Q: Let R be a ring and M be an R-module. Ther

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Q: Suppose that R and S are isomorphic rings. Prove that R[r] = S[r].

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Q: 2. Let Z[/2] = {a+b2 |a, b eZ} and let H= { a 2b : a,b eZ}. a. Show that Z[2] and H are isomorphic…

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

A: Since you have asked multiple question,as per our guidelines we are supposed to answer only one…

Q: Let R be a ring with unity e. Verify that the mapping θ: Z---------- R defined by θ (x) = x • e is a…

A: Let R be a ring with unity e, verify the mapping θ:Z→R defined by θx=x.e is a homomorphism If R and…