Let R be a commutative ring with identity with the property that every ideal in R...

Let R be a commutative ring with identity with the property that every ideal in R is principal. Prove that every homomorphic image of R has the same property.

Expert Solution

Colby Messinger answered 2 years ago

Let R be a commutative ring with unity. If I is a prime ideal of R,...

Let R be a commutative ring with unity. If I is a prime ideal of R, prove that I[x] is a prime ideal of R[x].

Given a commutative ring with identity R. A. Prove that a unit is not a zero...

Given a commutative ring with identity R. A. Prove that a unit is not a zero divisor. B. Prove that a zero divisor cannot be a unit.

Let R be a commutative domain, and let I be a prime ideal of R. (i)...

Let R be a commutative domain, and let I be a prime ideal of R. (i) Show that S defined as R \ I (the complement of I in R) is multiplicatively closed. (ii) By (i), we can construct the ring R1 = S-1R, as in the course. Let D = R / I. Show that the ideal of R1 generated by I, that is, IR1, is maximal, and R1 / I1R is isomorphic to the field of fractions of...

Let R and S be commutative rings with unity. (a) Let I be an ideal of...

Let R and S be commutative rings with unity. (a) Let I be an ideal of R and let J be an ideal of S. Prove that I × J = {(a, b) | a ∈ I, b ∈ J} is an ideal of R × S. (b) (Harder!) Let L be any ideal of R × S. Prove that there exists an ideal I of R and an ideal J of S such that L = I × J.

Let R be a ring (not necessarily commutative), and let X denote the set of two-sided...

Let R be a ring (not necessarily commutative), and let X denote the set of two-sided ideals of R. (i) Show that X is a poset with respect to to set-theoretic inclusion, ⊂. (ii) Show that with respect to the operations I ∩ J and I + J (candidates for meet and join; remember that I+J consists of the set of sums, {i + j} where i ∈ I and j ∈ J) respectively, X is a lattice. (iii) Give...

Let R be a commutative ring with unity. Prove that f(x) is R[x] is a unit...

Let R be a commutative ring with unity. Prove that f(x) is R[x] is a unit in R[x] iff f(x)=a is of degree 0 and is a unit in R.

10.3.6 Exercise: Product of Pairwise Comaximal Ideals. Let R be a commutative ring, and let {A1,...,An}...

10.3.6 Exercise: Product of Pairwise Comaximal Ideals. Let R be a commutative ring, and let {A1,...,An} be a pairwise comaximal set ofn ideals. Prove that A1 ···An = A1 ∩ ··· ∩ An. (Hint: recall that A1 ···An ⊆ A1 ∩···∩An from 8.3.8).

Let A be a commutative ring and F a field. Show that A is an algebra...

Let A be a commutative ring and F a field. Show that A is an algebra over F if and only if A contains (an isomorphic copy of) F as a subring.

Problem 1. Suppose that R is a commutative ring with addition “+” and multiplication “·”, and...

Problem 1. Suppose that R is a commutative ring with addition “+” and multiplication “·”, and that I a subset of R is an ideal in R. In other words, suppose that I is a subring of R such that (x is in I and y is in R) implies x · y is in I. Define the relation “~” on R by y ~ x if and only if y − x is in I, and assume for the...

Problem 2. Suppose that R is a commutative ring, and that R[X] := { (a_0, a_1,...

Problem 2. Suppose that R is a commutative ring, and that R[X] := { (a_0, a_1, a_2, ...)^T | a_i is in R, a_i not equal to 0 for only finitely many i} is the set of polynomials over R, where we have named one particular element X := (0, 1, 0, 0, . . .)T . Show that R[X] forms a commutative ring with a suitably-chosen addition and multiplication on R[X]. This will involve specifying a “zero” element of...

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