Let's enlarge the ring R, by adjoining a new element & such that ² = 0. In this way, we'll obtain a new ring R[ɛ]. (a) Explain why every element of R[s] can be written in the form a + be, where a, b E R. (b) Given two elements a + be €R[ɛ] and c + de € R[e], write down a formula for their product: (a + b)(c + de) Make use of the relation 2 = 0 to simplify your result. (c) Show that the element 1+ is a unit in R[ɛ], by finding its multiplicative inverse.
Let's enlarge the ring R, by adjoining a new element & such that ² = 0. In this way, we'll obtain a new ring R[ɛ]. (a) Explain why every element of R[s] can be written in the form a + be, where a, b E R. (b) Given two elements a + be €R[ɛ] and c + de € R[e], write down a formula for their product: (a + b)(c + de) Make use of the relation 2 = 0 to simplify your result. (c) Show that the element 1+ is a unit in R[ɛ], by finding its multiplicative inverse.
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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![2. Let's enlarge the ring R, by adjoining a new element & such that ɛ² = 0. In this way, we'll obtain a new
ring R[ɛ].
(a) Explain why every element of R[ɛ] can be written in the form a + bɛ, where a, b € R.
(b) Given two elements a + bɛ € R[ɛ] and c + dɛ € R[ɛ], write down a formula for their product:
(a + b)(c + dɛ)
Make use of the relation ² = 0 to simplify your result.
(c) Show that the element 1+ is a unit in R[ɛ], by finding its multiplicative inverse.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf4d5614-e5fa-4399-aabc-c345eeef0588%2F9fb503f0-83aa-4594-9135-bb89c1f2a1f0%2Fgmna743_processed.png&w=3840&q=75)
Transcribed Image Text:2. Let's enlarge the ring R, by adjoining a new element & such that ɛ² = 0. In this way, we'll obtain a new
ring R[ɛ].
(a) Explain why every element of R[ɛ] can be written in the form a + bɛ, where a, b € R.
(b) Given two elements a + bɛ € R[ɛ] and c + dɛ € R[ɛ], write down a formula for their product:
(a + b)(c + dɛ)
Make use of the relation ² = 0 to simplify your result.
(c) Show that the element 1+ is a unit in R[ɛ], by finding its multiplicative inverse.
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