Could you explain how to show 9.6 in detail? I included list of theorems and definitions from the textbook. **Problem 9.6:**Let $ R $ be a commutative ring. If \[ I = \{ a \in R : a^n = 0 \text{ for some } n \in \mathbb{N} \}, \] show that $ I $ is an ideal of $ R $.**Explanation:**This problem involves proving that a specific subset $ I $ of a commutative ring $ R $ is an ideal. The subset $ I $ consists of all elements $ a $ in $ R $ such that there exists a natural number $ n $ for which the power $ a^n $ equals zero.To show that $ I $ is an ideal, we must demonstrate two properties:1. **Additive Closure:** For any $ a, b \in I $, the sum $ a + b $ should also be in $ I $.2. **Absorption Property:** For any $ a \in I $ and any $ r \in R $, the product $ ra $ should be in $ I $.The goal of this exercise is to verify these properties using the definition of $ I $ and the properties of the commutative ring $ R $. **Definition 9.1.** Let $ R $ be a ring. Then a subring $ I $ of $ R $ is said to be an ideal if $ ir, ri \in I $ for all $ i \in I $ and $ r \in R $. We call this the absorption property.**Theorem 9.1.** Let $ R $ be a ring and $ I $ a subset of $ R $. Then $ I $ is an ideal if and only if1. $ 0 \in I $;2. $ i - j \in I $ for all $ i, j \in I $; and3. $ ir, ri \in I $ for all $ i \in I, r \in R $.**Example 9.1.** Let $ n $ be any integer. Then $ n\mathbb{Z} $ is an ideal of $ \mathbb{Z} $. Indeed, we already know that it is a subring. But also, if $ nk \in n\mathbb{Z} $, then for any integer $ r $, $ r(nk) = n(rk) \in n\mathbb{Z} $.**Example 9.2.** Let $ I $ be the set of all polynomials $ f(x) \in \mathbb{R}[x] $ such that $ f(0) = 0 $. We claim that $ I $ is an ideal in $ \mathbb{R}[x] $. Certainly $ I $ contains the zero polynomial. Also, if $ f(0) = g(0) = 0 $, then $ (f-g)(0) = f(0) - g(0) = 0 $, hence $ f(x) - g(x) \in I $. Also, if $ f(0) = 0 $ and $ h(x) \in \mathbb{R}[x] $, then $ h(0)f(0) = h(0)0 = 0 $. Hence, $ h(x)f(x) \in I $.**Example 9.3.** Let $ I $ be the set of all polynomials in \( \mathbb{Z}[x]

Name: Could you explain how to show 9.6 in detail? I included list of theore
Uploaded: 2024-03-20T06:38:44.000Z
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Description: Could you explain how to show 9.6 in detail? I included list of theorems and definitions from the textbook. **Problem 9.6:**Let $ R $ be a commutative ring. If \[ I = \{ a \in R : a^n = 0 \text{ for some } n \in \mathbb{N} \}, \] show that $ I $