27. For a commutative ring R with unity show that the relation a ~ b if a is an associate of b (that is, if a = bu for u a unit in R) is an equivalence relation on R.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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, then p is an irreducible.
26. Prove that if p is an irreducible in a UFD, then p is a prime.
27. For a commutative ring R with unity show that the relation a ~ b if a is an associate of b (that is, if a = bu
for u a unit in R) is an equivalence relation on R.
%3D
28. Let D be an integral domain. Exercise 37, Section 18 showed that (U, ·) is a group where U is the set of units
of D. Show that the set D* - U of nonunits of D excluding ) is closed uu
under the mul
1:
Transcribed Image Text:, then p is an irreducible. 26. Prove that if p is an irreducible in a UFD, then p is a prime. 27. For a commutative ring R with unity show that the relation a ~ b if a is an associate of b (that is, if a = bu for u a unit in R) is an equivalence relation on R. %3D 28. Let D be an integral domain. Exercise 37, Section 18 showed that (U, ·) is a group where U is the set of units of D. Show that the set D* - U of nonunits of D excluding ) is closed uu under the mul 1:
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