Is it true or false that if I is a principal ideal in a commutative ring R with multiplicative identity then R/I is an integral domain?

Q: Show that a unit of a ring divides every element of the ring.

A: Let us assume a unit in a ring R as a. Let us assume any element as x in that ring R.  By using the…

Q: If R= = b {(6) a. b. c real} and 1-{(5) b real). | a, I = (a) R is a ring. (b) I is an ideal of R.…

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Q: Complete the proof of Theorem (the ring of cosests) by proving the right distributive law in R/I.

A: Let I be an ideal of the ring R. Then the set RI of additive cosets r+I of I in R forms a ring with…

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

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Q: Let R be a commutative ring with unit which is not an integral domain. Show that quotient and…

A: Question asks to show that if R is a commutative ring with a unit that is not an integral domain,…

Q: Suppose that (R, +, . ) is a commutative ring with identity and it has nontrivial ideals, then R…

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Q: The zero ring ⟨{0}, #, ∗⟩ is a field with respect to addition # and multiplication ∗ because it…

A: The given Statement is  the Zero ring is a field .

Q: (b) (-a)b = −(ab) = a(−b) for all a,b € R.

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Q: Let R be a ring with unity and assume p, q, r ∈ R∗ Find (pqr)-1and also prove that it’s the inverse…

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Q: Show that in the factor ring Z[i] / , the element x + is a unit.

A: Given a factor ring Zx2x+1=a0+a1x,+⋯+2x+1 |a1∈Z  As 2x+1+2x+1=0+2x+1 Since,  2x+1∈2x+1   as aH=H iff…

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: Let T = { | EQ, a and b are relatively prime and 5 does not divide b}. Show that T is a ring under…

A: SOLUTION:  Let T=ab: ab∈ℚ,a and b are relatively prime and 5 does not divide b   We  have  show…

Q: Prove or disprove that Q(√3) and Q(√-3) are ring-isomorphic.

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Q: Let R be a commutative ring with identity 1R ≠ 0R. Prove that R is an integral domain if and only if…

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Q: Let (R, +, .) be a nontrivial ring with * identity, prove that 1 0

A: It is given that (R,+, .) be a nontrivial ring with identity. Now we have to show that 1≠0. So, (R,…

Q: Q2: Prove that the intersection of any two ideals of a ring R is also an ideal.

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Q: Show that a commutative ring R with unity is a field if and only if its only two ideals are (0) and…

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Q: ísomorp

A: Isomorphic to quotient ring:

Q: (†) \u,v≤U ([u‡ 0 and v‡0] → u • v‡ 0); (⋆) Vx,y,z€U ([z ‡ 0 and zx = zy] →x=y).

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Q: Verify that the ring R[x]/ is isomorphic to the ring of complex numbers.

A: First Isomorphism Theorem-  Let R and S be rings, and let f:R→S be a ring homomorphisms. Then if…

Q: Q) Let (H,+10, -10) be a subring of the ring (Z10, +10. -10 ), where H = {[0], [2], [4], [6], [8]}.…

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Q: Let P be a prime ideal and R a commutative ring with identity. If R/P is an integral domain, explain…

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Q: Is the factor ring Z5[x]/ a field?

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

A: The ring Z has identity 1 as 1·a=a·1=a∀a∈Z The ring 3Z has no identity i.e. there does not exist…