Exercise 3: Let R be a commutative ring. (a) Let I and J be ideals in R. Show that In J is also an ideal in R. (b) Now let R be a principal ideal domain. Let a, b E R and let (a) (b) = (m) for some me R. Show that m is the least common multiple of a and b, written m = lcm(a, b), meaning that a | m, bm, and if c is an element such that ac and b | c, then m | c. [Note that m is defined only up to multiplication by a unit, by Exercise 1.] (c) Still assuming that R is a PID, suppose that (a, b) = (1) = R. Show that (a) n (b) = (ab). (Hint: You can write 1 = au + bv for some u, v € R.)

Q: Let R (xy + yz, x2 – 3z?) be an ideal in the ring K[x, y, z] is R is radical ideal

A: Given: I(X) = (xy+yz, x^2-3z^2). Clearly, xy+yz, x^2-3z^2 vanish on X. Conversely, if a polynomial…

Q: Determine whether T={(a, - a) | aEZ} is a subring of ZxZ.

A:

Q: 18. Prove that in a Euclidean ring R, (a, b) can be found as follows : b= 90 a+ r,, where d (r) <d…

A:

Q: (15 points) Prove the ring Z[√−2] = {a+b√−2 | a,b ≤ Z} is a Euclidean domain with norm 8(a+b√2) = a²…

A:

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

A:

Q: Let M2(Z) be a ring and =(( ), a,b e Z}. Then S is ideal %D .of M2(Z) l choose one true Error

A:

Q: Let S = {a + bi : a, b e Z, b is even}. Then S is %3D not an ideal of Z[i] an ideal of Z[i]

A:

Q: Consider the ring R = 2Z and the ideal I = 24% of R. Does the ring R/I have an identity? Explain.

A: Please refer below page.If you have any doubts please feel free to ask.Explanation:Step 1: Step 2:

Q: 3 Consider the initial value problem y' 4t + 3et, y(0) = yo. y = 2 Find the value of yo that…

A:

Q: Let T = Z - 5Z. Show that ZT is a local ring. What is its unique maximal ideal?

A:

Q: Let A and В be ideals in principal ideal domain R , where A + R and В + R . Prove that the ideal AB…

A:

Q: (8) Show that for a field F, the set S of all all matrices of the form a b (88) 00 for a, b F is a…

A: An ideal is a special kind of subring of a ring R. If a∈I and r∈R implies that ra∈I then, the…

Q: a) The ring R[x, y]/(x + 1) is a field. b) The ring Z[x]/(7) is a principal ideal domain

A: a) The ring R[x, y]/(x + 1) is not a field. To determine if it is a field, we need to check if every…

Q: 4. Let f: RS be a ring homomorphsim, I an ideal in R, and J an ideal in S. (i) Prove that f-¹(J) is…

A:

Q: Let R be a ring. (a) Show that R has exactly one zero, 0R. (b) If R also has an identity, 1R, show…

A: Given: R is a ring.We need to prove the following:(a) Show that R has exactly one zero, 0R. (b) If R…

Q: (x), J = (x + 1) en K := (x²020 + x) in the polynomial ring R[x]. Rewrite the ideal I · (I + (J^ K))…

A: Aim: To define the principal ideals and in the polynomial ring . And to define as principal…

Q: Solve the D. E. to the I. V. P. sinx(e-y + 1)x' – cosx = 1, y(0) = 0 %3D %3D (1 – e') (1 + cos.x) =…

A: We will find out the required solution.

Q: Use Laplace transforms to solve the following initial value problem. y" – yf – 6y = 0; y(0) = 1 ,…

A: The given IVP is as follows. y''-y'-6y=0; y0=1, y'0=-1 Apply Laplace transform on the IVP as…

Q: Determine all the values of the parameter b € R, for which the polynomial f = x³y + bxy³2² belongs…

A: Given information:f(x,y,z)=x3y+bxy3z2Ideal: (x2+2y2,xz−y)To find: The suitable value of the…

Q: Let a ER which expressions are correct? and give the justification why they are part of the Borel…

A:

Q: Let I = {(a, 0) | a e Z}. Show that I is a prime ideal, but not a maximal ideal of the ring Z×Z.

A: Ideal of a ring

Q: 2. Z [x] is a principal ideal domain. a) True b) False O a) True O b) False

A:

Q: Let R be a commutative ring and let A be an ideal of R. Show that VA = {x € R:x" e A for some…

A:

Q: a) Let Ø: R – S be a homomorphism of rings R and S and let B be an ideal of S. Show that 01(B) =…

A:

Q: Prove that if (I,+,) is an ideal of the ring (R,+, ), then rad I In rad R. %3D

A: The term radical is used when we think about ideals and when we talk about ideals definitely…

Q: THE RADICAL OF AN IDEAL IS ALSO AN IDEAL.

A: We have to show that  RADICAL OF AN IDEAL IS AN IDEAL

Q: show by of R then IUI is not ideal example if In and Iz are icdeals an

A:

Q: 6. Let I and J be ideals in the ring R. (a) Prove that InJ is an ideal in R. (b) Prove that I+J =…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: Let an ideal I = {(a, 0): a ∈ Z} of the ring R = Z ⊕ Z be given. I, R is a prime Is it the ideal

A: To prove: I is an ideal and it is prime ideal of R. Here, R=Z⊕Z Let ,I=a,0:a∈ℤ A non empty subset I…

Q: Let (R, +,.) be any ring. Define Rx Z = {(r,n):r ER,n E Z} and (r,n) + (s, m) = (r + s,m + n) (r,n).…

A: Ring : A ring is a triplet R,+,·consists of a non-empty set R with two  binary operations of…

Q: 2- An example of two ideals A and B of the ring (Z12. +2-u) such thata B are: (a) A and B (c) A- and…

A: First we find all subgroup of Z12

Q: Define the following principal ideals of the polynomial ring R[x]: I := (x³ + 1), J := (x + 1) and K…

A: Suppose  that   be  a  polynomial ring and  I = and .Then, IJ=Here, given that   be the  polynomial…

Q: (3) Let R denote a commutative ring with a one. An element x of Ris termed nilpotent if and only 3n…

A: We now solve the Q3

Q: 2. In the ring (4Z, +,.), the ideal (8) is (a) not prime (b) maximal (c) maximal and not prime (d)…

A: