Question 5.Prove that the ring of integers Z₁, 0, can be embedded inevery integral domain D=D, +, of prime ring characteristic p € P.

Given that ℤp,⊕,⊙ is a ring of integers. Also D=D,+,· is an integral domain of prime characteristics…

Answered: Question 5. Prove that every integral…

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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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 $ℤ_{p} = \{1, 2, 3, \dots, p - 1\}$ , 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 $D = <D, +, \cdot>$ .