Construct the tables of this ring, write the elements in their most simple form, and determine the type of ring. .Z3 (x]/ (x² + x + 1)
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- Give an example where a and b are not zero divisors in a ring R, but the sum a + b is a zero divisor.If a and b are nilpotent element of a commutative ring, show that a+b is also nilpotent. Give an example to show that this may fail if the ring is not commutative( maybe using a counterexample (a+b)6=0)Need typed and correct solution otherwise will get down vote The characteristics of local rings is either 0 or prime.
- Enter the multiplicative inverse of the element [22]44 x + [17]44 in the ring ℤ44[x]. When you enter your answer, make sure that all congruence classes are in the form [a]44 with 0 ≤ a < 44. You don't have to type the brackets. For example, if your answer is [2]44 x2 + [3]44, you can type 2x^2+3.Give an example of ring elements a and b with the properties thatab = 0 but ba ≠ 0.Give all the associates of 2x − 7 in the ring (a) Z[x]; (b) Q[x]; and (c) Z_11[x].
- Which ring is fx=2x^2-1 irreducible?Z[x]Q[x]Z2[x]Z3[x]Which of the rings, if any, are fields? Select one: a. Z3 × Z b. Z10 c. {a + ib : a, b ∈ Q}In R=Z_24, partition this ring into its zero, zero divisors and units. (Draw the partition on your solutions.)What are the units in R? (List them all below in increasing order without spaces.) hb. 1,2,3For the previous item, what are the zero divisors in R? Write them all below in increasing order without spaces.