6.2.9. Show that if R is a ring with identity, then the principal ideal gener ated by x ERis
74. Let R be a commutative ring with identity such that not every ideal is a principal ideal principal ideal. (b) If I is the ideal of part (a), show that R/I is a principal ideal ring 74. Let R be a commutative ring with identity such that not every ideal is a principal ideal principal ideal. (b) If I is the ideal of part (a), show that R/I is a principal ideal ring
66. Let R be a commutative ring with identity. An ideal I of R is irreducible if it cannot be expressed as the intersection of two ideals of R neither of which is contained in the other. the following. (a) If P is a prime ideal then P is irreducible. (b) If z is a non-zero element of R, then there is an ideal I, maximal with respect to the property that r gI, and I is irreducible. (c) If...
Please answer all parts. Thank you! 20. Let R be a commutative ring with identity. We define a multiplicative subset of R to be a subset S such that 1 S and ab S if a, b E S. Define a relation ~ on R × S by (a, s) ~ (a, s') if there exists an s"e S such that s* (s,a-sa,) a. 0. Show that ~ is an equivalence relation on b. Let a/s denote the equivalence class...
117. If R is any ring with identity, let J(R) denote the Jacobson radical of R. Show that if e is any idempotent of R, then J(e Re) eJ(R)e. 117. If R is any ring with identity, let J(R) denote the Jacobson radical of R. Show that if e is any idempotent of R, then J(e Re) eJ(R)e.
1. Let R be a commutative ring with identity and let u e R be nilpotent elements a) (3 pt) Show that x + y and xy are nilpotent elements. b) (3 pt) Show that if u is a unit of R and t is nilpotent, then u is a umit. ) 3 pt) Show that if R is not commutative, neither of the above necessarily holds (r t y is not necessarily nilpotent and u 4- r is not...
Suppose R is a principal ideal domain, and let S be a multiplicatively closed subset of R not containing 0. Show that S-R is a principal ideal domain. Let I be an ideal of a principal ideal domain R. Show that R/I is a principal ideal domain if and only if I is prime.
Let R be a ring with identity 1. Suppose that 08 a € R satisfies a? = a. Show that for each TER, there exists a positive integer n such that [(1 – a)ral" = 0. What is the smallest possible value of n that works for all r ER?
Let R be a commutative ring with unity. If I is a prime ideal of R, prove that I[x] is a prime ideal of R[x].
6. If R is a primitive ring with identity and e ε R is such that e, e 0, then (a) eRe is a subring of R, with identity e. (b) eRe is primitive. [Hint: if R is isomorphic to a dense ring of endomorphisms of the vector space Vover a division ring D, then Ve is a D-vector space and eRe is isomorphic to a dense ring of endomorphisms of Ve.] 6. If R is a primitive ring with...
3. Consider the ring R- Zz[x] and the ideal Ixx+1>, (a) Is I a prime ideal? Is I a maximal ideal? (b) Find all the multiplicative units of R/I (a+ bx cx2 a, b, c E 2. Is the group of units cyclic? If so, give a generator. If not, determine to what commor group it is isomorphic?