Question

Let W = { 0, 1, 2, 3, 4, 5,6, 7,8,9 } be the set with addition and multiplication define by the tables. Assume that associativity and distributivity properties are hold. + | 0 1 2 3 4 5 6 7 8 9 0 | 0 1 2 3 4 5 6 7 8 9 1 | 1 2 3 4 5 6 7 8 9 1 2 | 2 3 4 5 6 7 8 9 1 2 3 | 3 4 5 6 7 8 9 1 2 3 4 | 4 5 6 7 8 9 1 2 3 4 5 | 5 6 7 8 9 1 2 3 4 5 6 | 6 7 8 9 1 2 3 4 5 6 7 | 7 8 9 1 2 3 4 5 6 7 8 | 8 9 1 2 3 4 5 6 7 8 9 | 9 1 2 3 4 5 6 7 8 9 x | 0 1 2 3 4 5 6 7 8 9 0 | 0 0 0 0 0 0 0 0 0 0 1 | 0 1 2 3 4 5 6 7 8 9 2 | 0 2 4 6 8 1 3 5 7 9 3 | 0 3 6 9 3 6 9 3 6 9 4 | 0 4 8 3 7 2 6 1 5 9 5 | 0 5 1 6 2 7 3 8 4 9 6 | 0 6 3 9 6 3 9 6 3 9 7 | 0 7 5 3 1 8 6 4 2 9 8 | 0 8 7 6 5 4 3 2 1 9 9 | 9 9 9 9 9 9 9 9 9 9 Then show that (i) W is a ring . (ii)Is W commutative ring with multiplicative identity? (iii) Find all the unit elements of W. (iv) Find all the nilpotent elements of W. (v) Find all the zero divisors of W. (vi) Find all the principal ideals in W. Also tell us which are the prime ideals and which are the maximal ideals? (vii) Find the nilradical of each prime ideal of W. (viii) For each principle ideal of W, find its radical ideal. (i) Show that T = {0, 4, 8, 12, 16} is a subring of the ring ? 20 or T is an ideal of ? 20 or T is both. (ii) Find the annihilator of the elements {4,7,9,11,15} ? ? 20 . (iii) Give the justifications that the following pairs of the rings are isomorphic or not to each other.

          Let W = { 0, 1, 2, 3, 4, 5,6, 7,8,9 } be the set with addition and multiplication define by the tables. Assume that associativity and distributivity properties are hold.

+ | 0 1 2 3 4 5 6 7 8 9
0 | 0 1 2 3 4 5 6 7 8 9
1 | 1 2 3 4 5 6 7 8 9 1
2 | 2 3 4 5 6 7 8 9 1 2
3 | 3 4 5 6 7 8 9 1 2 3
4 | 4 5 6 7 8 9 1 2 3 4
5 | 5 6 7 8 9 1 2 3 4 5
6 | 6 7 8 9 1 2 3 4 5 6
7 | 7 8 9 1 2 3 4 5 6 7
8 | 8 9 1 2 3 4 5 6 7 8
9 | 9 1 2 3 4 5 6 7 8 9

x | 0 1 2 3 4 5 6 7 8 9
0 | 0 0 0 0 0 0 0 0 0 0
1 | 0 1 2 3 4 5 6 7 8 9
2 | 0 2 4 6 8 1 3 5 7 9
3 | 0 3 6 9 3 6 9 3 6 9
4 | 0 4 8 3 7 2 6 1 5 9
5 | 0 5 1 6 2 7 3 8 4 9
6 | 0 6 3 9 6 3 9 6 3 9
7 | 0 7 5 3 1 8 6 4 2 9
8 | 0 8 7 6 5 4 3 2 1 9
9 | 9 9 9 9 9 9 9 9 9 9

Then show that
(i) W is a ring .
(ii)Is W commutative ring with multiplicative identity?
(iii) Find all the unit elements of W.
(iv) Find all the nilpotent elements of W.
(v) Find all the zero divisors of W.
(vi) Find all the principal ideals in W. Also tell us which are the prime ideals and which are the maximal ideals?
(vii) Find the nilradical of each prime ideal of W.
(viii) For each principle ideal of W, find its radical ideal.

(i) Show that T = {0, 4, 8, 12, 16} is a subring of the ring ? 20 or T is an ideal of ? 20 or T is both.
(ii) Find the annihilator of the elements {4,7,9,11,15} ? ? 20 .
(iii) Give the justifications that the following pairs of the rings are isomorphic or not to each other.

Let W = 0, 1, 2, 3, 4, 5,6, 7,8,9 be the set with addition and multiplication define by the tables. Assume that associativity and distributivity properties are hold.

+ | 0 1 2 3 4 5 6 7 8 9
0 | 0 1 2 3 4 5 6 7 8 9
1 | 1 2 3 4 5 6 7 8 9 1
2 | 2 3 4 5 6 7 8 9 1 2
3 | 3 4 5 6 7 8 9 1 2 3
4 | 4 5 6 7 8 9 1 2 3 4
5 | 5 6 7 8 9 1 2 3 4 5
6 | 6 7 8 9 1 2 3 4 5 6
7 | 7 8 9 1 2 3 4 5 6 7
8 | 8 9 1 2 3 4 5 6 7 8
9 | 9 1 2 3 4 5 6 7 8 9

x | 0 1 2 3 4 5 6 7 8 9
0 | 0 0 0 0 0 0 0 0 0 0
1 | 0 1 2 3 4 5 6 7 8 9
2 | 0 2 4 6 8 1 3 5 7 9
3 | 0 3 6 9 3 6 9 3 6 9
4 | 0 4 8 3 7 2 6 1 5 9
5 | 0 5 1 6 2 7 3 8 4 9
6 | 0 6 3 9 6 3 9 6 3 9
7 | 0 7 5 3 1 8 6 4 2 9
8 | 0 8 7 6 5 4 3 2 1 9
9 | 9 9 9 9 9 9 9 9 9 9

Then show that
(i) W is a ring .
(ii)Is W commutative ring with multiplicative identity?
(iii) Find all the unit elements of W.
(iv) Find all the nilpotent elements of W.
(v) Find all the zero divisors of W.
(vi) Find all the principal ideals in W. Also tell us which are the prime ideals and which are the maximal ideals?
(vii) Find the nilradical of each prime ideal of W.
(viii) For each principle ideal of W, find its radical ideal.

(i) Show that T = 0, 4, 8, 12, 16 is a subring of the ring ? 20 or T is an ideal of ? 20 or T is both.
(ii) Find the annihilator of the elements 4,7,9,11,15 ? ? 20 .
(iii) Give the justifications that the following pairs of the rings are isomorphic or not to each other.

Added by Sofia B.

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