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7. The following are equivalent: (a) R is prime; (6) ab € R and aRb =...
3. Let I be a left ideal of R and let (: R) reRRCI (a) : R) is an ideal of R. If l is regular, th...
3. Let I be a left ideal of R and let (: R) reRRCI (a) : R) is an ideal of R. If l is regular, then (: R) is the largest ideal of R that is contained in 1 (b) If I is a regular maximal left ideal of Rand AR/I, then (A(R). Therefore J(R) na:R), where /runs over all the regular maximal left ideals of R. Theorem 1.4. Let B be a subset of a left module A...
Let R be an ED but not a field, with a norm function N R-Z U0 such that N(ab for all a, bE R. (a)...
Let R be an ED but not a field, with a norm function N R-Z U0 such that N(ab for all a, bE R. (a) Prove that N(1R) (b) If r E R is a unit, show that N(r) 1. (c) If r E R is nonzero, show that N(r) 0. (d) For any r E R, prove that if N(r) 1, then r is a unit N(a)N(b) e) For any r e R if N() is a prime mumber,...
1. (2 marks) Let S 2,3,4,5,6,7,8,9, 10, 11, 12). Let r be the relation on the set S defined as follows: Va,bE S, arb if and only if every prime number that divides a is a factor of b and a S b. The r...
1. (2 marks) Let S 2,3,4,5,6,7,8,9, 10, 11, 12). Let r be the relation on the set S defined as follows: Va,bE S, arb if and only if every prime number that divides a is a factor of b and a S b. The relation T is a partial order relation (you do not need to prove this). Draw the Hasse diagram for T
1. (2 marks) Let S 2,3,4,5,6,7,8,9, 10, 11, 12). Let r be the relation on the...
(7) Let R be a ring with 1 and let M be a unital left R-module....
(7) Let R be a ring with 1 and let M be a unital left R-module. If I is a right ideal of R then the annihilator of I in M is defined to be AnnM(I) = {m € M: am=0 for all a € 1}. (a) Prove that Annm(I) is a submodule of M. (b) Take R = Z and M = Z/3Z Z/102 x Z/4Z. If I = 2Z describe AnnM(I) as a direct product of cyclic groups.
5. An elementa of a ring Ris regular in the sense of Von Neumann) if there...
5. An elementa of a ring Ris regular in the sense of Von Neumann) if there exists ER such that axa - a. If every element of R is regular, then R is said to be a regular ring. 3. SEMISIMPLE RINGS (a) Every division ring is regular. (b) A finite direct product of regular rings is regular. (c) Every regular ring is semisimple. The converse is false (for example, Z). (d) The ring of all lincar transformations on a...
66. Let R be a commutative ring with identity. An ideal I of R is irreducible if it cannot be expressed as the i...
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...
Every ring in this test is commutative with 1 and 1 0 1. Which of the followings are prime ideals of Z? (Separate your answers by commas.) A. ( B. (2). C. (9). D. (111). E. (101) 2. Which of the foll...
Every ring in this test is commutative with 1 and 1 0 1. Which of the followings are prime ideals of Z? (Separate your answers by commas.) A. ( B. (2). C. (9). D. (111). E. (101) 2. Which of the followings are ring homomorphisms? (Separate your answers by commas.) A.φ: Z → Z, defined by (n) =-n for all n E Z B. ф: Z[x] Z, defined by ф(p(z)) p(0) for all p(z) E Z[2] C. : C C....
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...
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...
Thanks 6. Let R be a ring and a € R. Prove that (i) {x E...
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6. Let R be a ring and a € R. Prove that (i) {x E R | ax = 0} is a right ideal of R (ii) {Y E R | ya=0} is a left ideal of R (iii) if L is a left ideal of R, then {z E R za = 0 Vae L} is a two-sided ideal of R NB: first show that each set in 6.(i), (ii), (iii) above is a subring T ool of...
Let R(A,B,C,D,E) be a relation with FDs F = {AB-C, CD-E, E–B} (2 Points) Select one:...
Let R(A,B,C,D,E) be a relation with FDs F = {AB-C, CD-E, E–B} (2 Points) Select one: O Ris in 3NF but not in BCNF. O Ris not in 3NF but in BCNF. O Ris in 3NF and in BCNF. R is not in 3NF and not in BCNF.
3. Let I be a left ideal of R and let (: R) reRRCI (a) : R) is an ideal of R. If l is regular, then (: R) is the largest ideal of R that is contained in 1 (b) If I is a regular maximal left ideal of Rand AR/I, then (A(R). Therefore J(R) na:R), where /runs over all the regular maximal left ideals of R. Theorem 1.4. Let B be a subset of a left module A...
Let R be an ED but not a field, with a norm function N R-Z U0 such that N(ab for all a, bE R. (a) Prove that N(1R) (b) If r E R is a unit, show that N(r) 1. (c) If r E R is nonzero, show that N(r) 0. (d) For any r E R, prove that if N(r) 1, then r is a unit N(a)N(b) e) For any r e R if N() is a prime mumber,...
1. (2 marks) Let S 2,3,4,5,6,7,8,9, 10, 11, 12). Let r be the relation on the set S defined as follows: Va,bE S, arb if and only if every prime number that divides a is a factor of b and a S b. The relation T is a partial order relation (you do not need to prove this). Draw the Hasse diagram for T
1. (2 marks) Let S 2,3,4,5,6,7,8,9, 10, 11, 12). Let r be the relation on the...
(7) Let R be a ring with 1 and let M be a unital left R-module. If I is a right ideal of R then the annihilator of I in M is defined to be AnnM(I) = {m € M: am=0 for all a € 1}. (a) Prove that Annm(I) is a submodule of M. (b) Take R = Z and M = Z/3Z Z/102 x Z/4Z. If I = 2Z describe AnnM(I) as a direct product of cyclic groups.
5. An elementa of a ring Ris regular in the sense of Von Neumann) if there exists ER such that axa - a. If every element of R is regular, then R is said to be a regular ring. 3. SEMISIMPLE RINGS (a) Every division ring is regular. (b) A finite direct product of regular rings is regular. (c) Every regular ring is semisimple. The converse is false (for example, Z). (d) The ring of all lincar transformations on a...
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...
Every ring in this test is commutative with 1 and 1 0 1. Which of the followings are prime ideals of Z? (Separate your answers by commas.) A. ( B. (2). C. (9). D. (111). E. (101) 2. Which of the followings are ring homomorphisms? (Separate your answers by commas.) A.φ: Z → Z, defined by (n) =-n for all n E Z B. ф: Z[x] Z, defined by ф(p(z)) p(0) for all p(z) E Z[2] C. : C C....
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...
Thanks
6. Let R be a ring and a € R. Prove that (i) {x E R | ax = 0} is a right ideal of R (ii) {Y E R | ya=0} is a left ideal of R (iii) if L is a left ideal of R, then {z E R za = 0 Vae L} is a two-sided ideal of R NB: first show that each set in 6.(i), (ii), (iii) above is a subring T ool of...
Let R(A,B,C,D,E) be a relation with FDs F = {AB-C, CD-E, E–B} (2 Points) Select one: O Ris in 3NF but not in BCNF. O Ris not in 3NF but in BCNF. O Ris in 3NF and in BCNF. R is not in 3NF and not in BCNF.
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