Answered: Prove that the ring of integers ≺…

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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Question

Prove that the ring of integers ≺ Zp,⊕,⊙ ≻ can be embedded in every integral domain D = ≺ D, +, · ≻ of prime ring characteristic p ∈ P.

Expert Solution

Step 1

Given that $<ℤ_{p}, \oplus, ⊙>$ is a ring of integers. Also $D = <D, +, \cdot>$ is an integral domain of prime characteristics $p \in ℙ$ .

We know that a ring $<R, +, \cdot>$ is embedded into a ring $<R', +', \cdot'>$ , if there is a subring $<S, +', \cdot'>$ of $<R', +', \cdot'>$ such that $<S', +', \cdot'> ≅ <R, +, \cdot>$

Since $ℤ_{p} = \{0, 1, 2, \dots, p - 1\}$ , therefore $<ℤ_{p}, \oplus, ⊙>$ is a ring with integer addition and multiplication modulo $p$ .

Since $D = <D, +, \cdot>$ is an integral domain of prime characteristic $p \in ℙ$ , therefore $p$ is the least positive integer such that $\underset{p times}{\underset{⏟}{e + e + \dots + e} = 0}$ where $e$ is multiplicative identity element of $D = <D, +, \cdot>$ and $0_{d}$ is an additive identity of $D$ .