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Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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**Theorem 2**

\((R/I, \oplus, \odot)\) is a ring (called the quotient ring).

**Proof** — Homework

1. \(\oplus\) is associative and commutative.
   - \(I\) is the zero element.
   - The opposite of \(a + I\) is \(-a + I\).

2. \(\odot\) is associative.
3. Distributive property applies.

This section introduces the concept of a quotient ring, which is defined using the notation \((R/I, \oplus, \odot)\). The proof that this structure forms a ring is to be completed as homework. The properties mentioned include associativity and commutativity for the addition operation (\(\oplus\)), with \(I\) being the zero element, and the existence of additive inverses. The multiplication operation (\(\odot\)) is associative, and the distributive property holds.
Transcribed Image Text:**Theorem 2** \((R/I, \oplus, \odot)\) is a ring (called the quotient ring). **Proof** — Homework 1. \(\oplus\) is associative and commutative. - \(I\) is the zero element. - The opposite of \(a + I\) is \(-a + I\). 2. \(\odot\) is associative. 3. Distributive property applies. This section introduces the concept of a quotient ring, which is defined using the notation \((R/I, \oplus, \odot)\). The proof that this structure forms a ring is to be completed as homework. The properties mentioned include associativity and commutativity for the addition operation (\(\oplus\)), with \(I\) being the zero element, and the existence of additive inverses. The multiplication operation (\(\odot\)) is associative, and the distributive property holds.
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