Let R be a ring with unity 1, and let u ∈ R be a unit. Let ne be a positive integer so that nu = 0. Show that
na = 0 for all a ∈ R.
2. Let R be a commutative ring with unity 1, and let a be a unit in R Let / be an ideal in R that contains the element a. Prove that / cannot be a proper ideal of R. 3. Let R be a commutative ring with unity 1 of order 30, and let be a prime ideal of R. Prove that is a maximal ideal of R
let R be a ring with unity (180) and let UE U(R). Suppose that I is an ideal of R sot. UE Io prove that I =R
Let R be a commutative ring with unity 1 and let I be a minimal ideal in R i.e. a nonzero ideal which does not properly contain another non-zero ideal. Show that either the product of two elements in I is always zero or there is an element in I that serves as unity in the ring I. Show also that in the latter case I is a field.
6. Let R be a non-trivial ring with unity. Suppose x E R satisfies x2 = 0. Show that x – 1 and x +1 are units in R.
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...
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?
(5) Suppose R is a commutative ring with unity, and r e R. Let A(r) {s E R : rs-0). Prove that A(r) is an ideal of R.
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].
Let R be a ring, let S be a subring of R and let' be an ideal of R. Note that I have proved that (5+1)/1 = {5 +1 | 5 € S) and I defined $:(5+1) ► S(SO ) by the formula: 0/5 + 1)=5+(SNI). In the previous video I showed that was well-defined. Now show that is a ring homomorphism. In other words, show that preserves both ring addition and ring multiplication. Then turn your work into this...
(i) Show that R is a subring of the polynomial ring Rx. | R{]4 (ii) Let k be a fixed positive integer and be the set of all polynomials of degree less than or equal to k. Is R[xk a subring of R[a]? 2r4+3x - 5 when it is (iii) Find the quotient q(x divided by P2(x) of the polynomial P1( and remainder r(x) - 2c + 1 in - (iv) List all the polynomials of degree 3 in Z...