Muslim Q2: (a) Let I be an ideal of the ring R, prove that f(I) is an ideal of R', when f is onto homomorphism from R into R'. (b) Prove that the ring of real numbers is embedding in the ring of the complex number. ****
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- Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .Prove Corollary 8.18: A polynomial of positive degree over the field has at most distinct zeros in
- Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.11. Show that defined by is not a homomorphism.a. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the characteristic of R. b. State the characteristic of Zn[ x ]. c. State the characteristic of Z[ x ].
- Prove the Unique Factorization Theorem in (Theorem). Theorem Unique Factorisation Theorem Every polynomial of positive degree over the field can be expressed as a product of its leading coefficient and a finite number of monic irreducible polynomials over . This factorization is unique except for the order of the factors.33. An element of a ring is called nilpotent if for some positive integer . Show that the set of all nilpotent elements in a commutative ring forms an ideal of . (This ideal is called the radical of .)Let I be an ideal in a ring R with unity. Prove that if I contains an element a that has a multiplicative inverse, then I=R.
- Use Theorem to show that each of the following polynomials is irreducible over the field of rational numbers. Theorem Irreducibility of in Suppose is a polynomial of positive degree with integral coefficients and is a prime integer that does not divide. Let Where for If is irreducible in then is irreducible in .12. Consider the mapping defined by . Decide whether is a homomorphism, and justify your decision.11. a. Give an example of a ring of characteristic 4, and elements in such that b. Give an example of a noncommutative ring with characteristic 4, and elements in such that .