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Chapter 6: Q9-b (page 149)

Let R be a ring with identity and let I be an ideal in R .
(b) If I contains a unit, prove that I = R .

Short Answer

Expert verified

It is proved that as $a^{- 1} a = 1_{R}$ , then I = R .

Step by step solution

Ideal definition

By the ideal definition, if $I \subseteq R$ , then $R \subseteq I$ .

Show that I = R

Consider that I is an ideal unit a.

Since a is the unit, then $a^{- 1} \in R$ . Also, if I is an ideal, $a^{- 1} a \in I$ .

Since $a^{- 1} a = 1_{R}$ , then I = R .

Hence, it is proved that I = R .

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Most popular questions from this chapter

(The Second Isomorphism Theorem) Let I and J be ideals in a ring R. Then $I \cap J$ is an ideal in I , and Jis an ideal in I + J by Exercises 19 and 20 of Section 6.1. Prove that $\frac{I}{I \cap J} ≅ \frac{I + J}{J}$ .[Hint: Show that $f : I \to \frac{I + J}{J}$ given by $f (a) = a + j$ is a surjective homomorphism with kernel $I \cap J$ .]

Question: Let be an $l$ ideal in a ring $R$ . Prove that every element in $\frac{R}{l}$ has a square root if and only if for every, $a \in R$ , there exists $b \in R$ such that $a - b^{2} \in l$ .

Let I be an ideal in R. Prove that K is an ideal, where $K = {a \in R / r a = I f o r e v e r y r \in R}$ .

Let $J$ be the set of all polynomials with zero constant term in $ℤ [x]$ .

(a) Show that $J$ is the principal ideal $(x)$ in $ℤ [x]$ .

Let $F$ be a field. Prove that every ideal in $F [x]$ is principal. [Hint: Use the Division Algorithm to show that the nonzero ideal $I$ in $F [x]$ is $(p (x))$ , where $p (x)$ is a polynomial of smallest possible degree in I ].

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Let R be a ring with identity and let I be an ideal in R .
(b) If I contains a unit, prove that I = R .

Short Answer

Step by step solution

Ideal definition

Show that I = R

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Most popular questions from this chapter

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Let R be a ring with identity and let I be an ideal in R .(b) If I contains a unit, prove that I = R .

Short Answer

Step by step solution

Ideal definition

Show that I = R

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Logic and Functions

Theoretical and Mathematical Physics

Probability and Statistics

Calculus

Pure Maths

Study anywhere. Anytime. Across all devices.

Let R be a ring with identity and let I be an ideal in R .
(b) If I contains a unit, prove that I = R .