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Consider the proof below of the fact that I + a = 1 +b iff a – bEI where R is a commutative right with unity and I is an ideal. Identify which "+" signs are notation and which "+" signs are operations. (=) Assume I + a = I + b. : a E l + a : a el + b . a = i+b for some i EI i = a – b : a – bel (=) Assume a – bE I Assume x E 1 + a :x = i + a for some i E I :x = i+ a + (-b+ b) :x = (i + a - b) + b i x E I + b : I + a CI+ b Assume x E I+b :x = i + b for some i El :x = i +b+ (-a+ b) : x = (i + b – a) + a : x €l + a : I+b SI+a ::I+ a = I + b

Consider the proof below of the fact that I + a = 1 +b iff a – bEI where R is a commutative right with unity and I is an ideal. Identify which "+" signs are notation and which "+" signs are operations. (=) Assume I + a = I + b. : a E l + a : a el + b . a = i+b for some i EI i = a – b : a – bel (=) Assume a – bE I Assume x E 1 + a :x = i + a for some i E I :x = i+ a + (-b+ b) :x = (i + a - b) + b i x E I + b : I + a CI+ b Assume x E I+b :x = i + b for some i El :x = i +b+ (-a+ b) : x = (i + b – a) + a : x €l + a : I+b SI+a ::I+ a = I + b

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Understanding Ideals in Commutative Rings: A Proof Analysis** Consider the proof below of the fact that \( I + a = I + b \) if and only if \( a - b \in I \) where \( R \) is a commutative ring with unity and \( I \) is an ideal. Identify which “+” signs are notation and which “+” signs are operations. **(\(\Rightarrow\)) Assume \( I + a = I + b \).** 1. \( a \in I + a \) \(\therefore a \in I + b\) 2. \(\therefore a = i + b\) for some \( i \in I \) 3. \(\therefore i = a - b\) 4. \(\therefore a - b \in I\) **(\(\Leftarrow\)) Assume \( a - b \in I\).** Assume \( x \in I + a \). 1. \(\therefore x = i + a\) for some \( i \in I \) 2. \(\therefore x = i + a + (-b + b)\) 3. \(\therefore x = (i + a - b) + b\) 4. \(\therefore x \in I + b\) 5. \(\therefore I + a \subseteq I + b\) Assume \( x \in I + b\). 1. \(\therefore x = i + b\) for some \( i \in I \) 2. \(\therefore x = i + b + (-a + a)\) 3. \(\therefore x = (i + b - a) + a\) 4. \(\therefore x \in I + a\) 5. \(\therefore I + b \subseteq I + a\) Thus, \( I + a = I + b \). Consider the proof above and justify each claim. --- This proof carefully uses properties of ideals in a commutative ring to show equivalences and membership in different sets defined by operations involving elements of the ring.
Transcribed Image Text:**Understanding Ideals in Commutative Rings: A Proof Analysis** Consider the proof below of the fact that \( I + a = I + b \) if and only if \( a - b \in I \) where \( R \) is a commutative ring with unity and \( I \) is an ideal. Identify which “+” signs are notation and which “+” signs are operations. **(\(\Rightarrow\)) Assume \( I + a = I + b \).** 1. \( a \in I + a \) \(\therefore a \in I + b\) 2. \(\therefore a = i + b\) for some \( i \in I \) 3. \(\therefore i = a - b\) 4. \(\therefore a - b \in I\) **(\(\Leftarrow\)) Assume \( a - b \in I\).** Assume \( x \in I + a \). 1. \(\therefore x = i + a\) for some \( i \in I \) 2. \(\therefore x = i + a + (-b + b)\) 3. \(\therefore x = (i + a - b) + b\) 4. \(\therefore x \in I + b\) 5. \(\therefore I + a \subseteq I + b\) Assume \( x \in I + b\). 1. \(\therefore x = i + b\) for some \( i \in I \) 2. \(\therefore x = i + b + (-a + a)\) 3. \(\therefore x = (i + b - a) + a\) 4. \(\therefore x \in I + a\) 5. \(\therefore I + b \subseteq I + a\) Thus, \( I + a = I + b \). Consider the proof above and justify each claim. --- This proof carefully uses properties of ideals in a commutative ring to show equivalences and membership in different sets defined by operations involving elements of the ring.
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