To prove: S 1 = { [ 0 ] , [ 3 ] , [ 6 ] } is a subring of ℤ 9 , and S 2 = { [ 0 ] , [ 5 ] } is a subring of ℤ 10 .

Question

Want to see the full answer?

Answer 1

Question

Chapter 5.1, Problem 55E

(a)

To determine

To prove: S1={[0],[3],[6]} is a subring of ℤ9, and S2={[0],[5]} is a subring of ℤ10.

(b)

To determine

The elements in S1⊕S2 and construct addition and multiplication tables for this ring.

(c)

To determine

Whether S1⊕S2 a commutative ring.

(d)

To determine

Whether unity element in S1⊕S2 exists.

(e)

To determine

All units in S1⊕S2, if they exist.

(f)

To determine

All zero divisors in S1⊕S2, if they exist.

(g)

To determine

All idempotent elements in S1⊕S2, if they exist.

(h)

To determine

All nilpotent elements in S1⊕S2, if they exist.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Students have asked these similar questions

Q1lal Let X be an arbitrary infinite set and let r the family of all subsets F of X which do not contain a particular point x, EX and the complements F of all finite subsets F of X show that (X.r) is a topology. bl The nbhd system N(x) at x in a topological space X has the following properties NO- N(x) for any xX N1- If N EN(x) then x€N N2- If NEN(x), NCM then MeN(x) N3- If NEN(x), MEN(x) then NOMEN(x) N4- If N = N(x) then 3M = N(x) such that MCN then MeN(y) for any уем Show that there exist a unique topology τ on X. Q2\a\let (X,r) be the topology space and BST show that ẞ is base for a topology on X iff for any G open set xEG then there exist A Eẞ such that x E ACG. b\Let ẞ is a collection of open sets in X show that is base for a topology on X iff for each xex the collection B, (BEB\xEB) is is a nbhd base at x. - Q31 Choose only two: al Let A be a subspace of a space X show that FCA is closed iff F KOA, K is closed set in X. الرياضيات b\ Let X and Y be two topological space and f:X -…

Q1\ Let X be a topological space and let Int be the interior operation defined on P(X) such that 1₁.Int(X) = X 12. Int (A) CA for each A = P(X) 13. Int (int (A) = Int (A) for each A = P(X) 14. Int (An B) = Int(A) n Int (B) for each A, B = P(X) 15. A is open iff Int (A) = A Show that there exist a unique topology T on X. Q2\ Let X be a topological space and suppose that a nbhd base has been fixed at each x E X and A SCX show that A open iff A contains a basic nbdh of each its point Q3\ Let X be a topological space and and A CX show that A closed set iff every limit point of A is in A. A'S A ACA Q4\ If ẞ is a collection of open sets in X show that ẞ is a base for a topology on X iff for each x E X then ẞx = {BE B|x E B} is a nbhd base at x. Q5\ If A subspace of a topological space X, if x Є A show that V is nbhd of x in A iff V = Un A where U is nbdh of x in X.

+ Theorem: Let be a function from a topological space (X,T) on to a non-empty set y then is a quotient map iff vesy if f(B) is closed in X then & is >Y. ie Bclosed in bp closed in the quotient topology induced by f iff (B) is closed in x- التاريخ Acy الموضوع : Theorem:- IP & and I are topological space and fix sy is continuous او function and either open or closed then the topology Cony is the quatient topology p proof: Theorem: Lety have the quotient topology induced by map f of X onto y. The-x: then an arbirary map g:y 7 is continuous 7. iff gof: x > z is "g of continuous Continuous function f

Answer 2

Question

Chapter 5.1, Problem 55E

(a)

To determine

To prove: S1={[0],[3],[6]} is a subring of ℤ9, and S2={[0],[5]} is a subring of ℤ10.

(b)

To determine

The elements in S1⊕S2 and construct addition and multiplication tables for this ring.

(c)

To determine

Whether S1⊕S2 a commutative ring.

(d)

To determine

Whether unity element in S1⊕S2 exists.

(e)

To determine

All units in S1⊕S2, if they exist.

(f)

To determine

All zero divisors in S1⊕S2, if they exist.

(g)

To determine

All idempotent elements in S1⊕S2, if they exist.

(h)

To determine

All nilpotent elements in S1⊕S2, if they exist.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 3

Question

Chapter 5.1, Problem 55E

(a)

To determine

To prove: S1={[0],[3],[6]} is a subring of ℤ9, and S2={[0],[5]} is a subring of ℤ10.

(b)

To determine

The elements in S1⊕S2 and construct addition and multiplication tables for this ring.

(c)

To determine

Whether S1⊕S2 a commutative ring.

(d)

To determine

Whether unity element in S1⊕S2 exists.

(e)

To determine

All units in S1⊕S2, if they exist.

(f)

To determine

All zero divisors in S1⊕S2, if they exist.

(g)

To determine

All idempotent elements in S1⊕S2, if they exist.

(h)

To determine

All nilpotent elements in S1⊕S2, if they exist.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 4

Question

Chapter 5.1, Problem 55E

(a)

To determine

To prove: S1={[0],[3],[6]} is a subring of ℤ9, and S2={[0],[5]} is a subring of ℤ10.

(b)

To determine

The elements in S1⊕S2 and construct addition and multiplication tables for this ring.

(c)

To determine

Whether S1⊕S2 a commutative ring.

(d)

To determine

Whether unity element in S1⊕S2 exists.

(e)

To determine

All units in S1⊕S2, if they exist.

(f)

To determine

All zero divisors in S1⊕S2, if they exist.

(g)

To determine

All idempotent elements in S1⊕S2, if they exist.

(h)

To determine

All nilpotent elements in S1⊕S2, if they exist.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 5

Chapter 5.1, Problem 55E

(a)

To determine

To prove: S1={[0],[3],[6]} is a subring of ℤ9, and S2={[0],[5]} is a subring of ℤ10.

(b)

To determine

The elements in S1⊕S2 and construct addition and multiplication tables for this ring.

(c)

To determine

Whether S1⊕S2 a commutative ring.

(d)

To determine

Whether unity element in S1⊕S2 exists.

(e)

To determine

All units in S1⊕S2, if they exist.

(f)

To determine

All zero divisors in S1⊕S2, if they exist.

(g)

To determine

All idempotent elements in S1⊕S2, if they exist.

(h)

To determine

All nilpotent elements in S1⊕S2, if they exist.

Answer 6

Chapter 5.1, Problem 55E

(a)

To determine

To prove: S1={[0],[3],[6]} is a subring of ℤ9, and S2={[0],[5]} is a subring of ℤ10.

Answer 7

Chapter 5.1, Problem 55E

(a)

To determine

To prove: S1={[0],[3],[6]} is a subring of ℤ9, and S2={[0],[5]} is a subring of ℤ10.

Answer 8

(b)

To determine

The elements in S1⊕S2 and construct addition and multiplication tables for this ring.

Answer 9

(b)

To determine

The elements in S1⊕S2 and construct addition and multiplication tables for this ring.

Answer 10

(c)

To determine

Whether S1⊕S2 a commutative ring.

Answer 11

(c)

To determine

Whether S1⊕S2 a commutative ring.

Answer 12

(d)

To determine

Whether unity element in S1⊕S2 exists.

Answer 13

(d)

To determine

Whether unity element in S1⊕S2 exists.

Answer 14

(e)

To determine

All units in S1⊕S2, if they exist.

Answer 15

(e)

To determine

All units in S1⊕S2, if they exist.

Answer 16

(f)

To determine

All zero divisors in S1⊕S2, if they exist.

Answer 17

(f)

To determine

All zero divisors in S1⊕S2, if they exist.

Answer 18

(g)

To determine

All idempotent elements in S1⊕S2, if they exist.

Answer 19

(g)

To determine

All idempotent elements in S1⊕S2, if they exist.

Answer 20

(h)

To determine

All nilpotent elements in S1⊕S2, if they exist.

Answer 21

(h)

To determine

All nilpotent elements in S1⊕S2, if they exist.

Answer 22

Expert Solution & Answer

Answer 23

Expert Solution & Answer

Answer 24

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 25

To prove: S 1 = { [ 0 ] , [ 3 ] , [ 6 ] } is a subring of ℤ 9 , and S 2 = { [ 0 ] , [ 5 ] } is a subring of ℤ 10 .

Solution Summary: The author explains that S_1 is a non-empty set.

Videos

To prove: S1={[0],[3],[6]} is a subring of ℤ9, and S2={[0],[5]} is a subring of ℤ10.

The elements in S1⊕S2 and construct addition and multiplication tables for this ring.

Whether S1⊕S2 a commutative ring.

Whether unity element in S1⊕S2 exists.

All units in S1⊕S2, if they exist.

All zero divisors in S1⊕S2, if they exist.

All idempotent elements in S1⊕S2, if they exist.

All nilpotent elements in S1⊕S2, if they exist.

Want to see the full answer?

Chapter 5 Solutions