Suppose phi maps R to S is a ring homomorphism and the image of of phi is not {0}. If R has a unity and S is an integral domain, show that phi carries the unity of R to the unity of  S.

Q: Let R be a ring with unity 1 and char (R) = 4. %3D Then R contains a subring isomorphic to Q ZO Z3 O

A: IN the given question,  Given that: R is a ring with unity 1 and char(R)=4. we have to find: we have…

Q: IF J is nil left ideal in an Artinian ring R, then J is

A:

Q: Show that the polynomial ring Z4 [x] over the ring Z₁ is infinite, but Z₁ [x] is of finite…

A:

Q: Let be a commutative ring with unity of characteristic 3. Compute and simplify (a + b)° for all…

A:

Q: Let R be a commutative ring. Then R is an integral domain if and only if ab= ac implies that b = c,…

A:

Q: Let ø : R  → R’  be a ring homomorphism and let N be an ideal of R.  Let N’ be an ideal either of ø…

A: Ideal: A non-empty subset I of a ring R is said to be ideal in a ring I if it satiesfies following…

Q: Let R be a commutative ring with unity, and let I be a proper idealwith the property that every…

A:

Q: Let R be a ring with a finite number n of elements. Show that the characteristic of R divides n.

A: Given :  R is a ring with n elements. To prove :  The characteristic of a ring R divides the number…

Q: Let R be a ring with unity 1. Show that S = {n· 1 | nE Z} is a sub- ring of R.

A:

Q: If Ø: R S is a ring homomorphism. The Ø preserves:

A:

Q: Suppose that Φ: R --> S is a ring homomorphism and that the imageof Φ is not {0}. If R has a…

A:

Q: Let R be a ring with unity 1 and char (R) = 3. Then R contains a subring isomorphic to

A: Let R be a ring with unity 1 and char(R)=3.  Then R contains a subring isomorphic to_______.

Q: 3. Prove that an ideal I in a ring R is the whole ring if and only if 1 e I.

A: Question: Prove that an ideal I in a ring R is the whole ring if and only if 1∈I. Proof: We have to…

Q: (3) Let A be commutative ring with identity, then A has just trivial ideals iff A is ....... O…

A: Here you have posted multiple question, So as per the policy I can answer only first question for…

Q: Indicate such a subring of the ring P[x] that contains P and is different from P but is not…

A: There are so many examples can be found.

Q: Suppose that (R, +, . ) is a commutative ring with identity and it has nontrivial ideals, then R…

A:

Q: consider the mapping from M2(z) into Z,prove or disprove that this is a ring homomorphism

A:

Q: Let R be a ring with unity 1R and let S be a subring of R containing 1R. If r∈R is a unit of R and…

A: Let R be a ring with unity 1R and let S be a subring of R containing 1R. If r∈R is a unit of R and…

Q: Let R be a ring and S be a subring of R with OS, OR being the zero elements in S, R respectively.…

A:

Q: If Ø: R → S is a ring homomorphism. The Ø preserves: All of these Nilpotent elements Units O…

A:

Q: Let R be a ring with identity 18 Show that the map Ø: ZRgive by Ø(n) = nx1R. is a ring homomorphism.…

A: Consider the provided question,

Q: determine all ring homomorphism from Z20 to Z30

A:

Q: 21. If R is a ring, prove that R[x] ~R, is the ideal generated by x.

A:

Q: Let R be a commutative ring with unity and let N= {a e R| a" = 0 for some n e Z"; n> 1} Show that N…

A:

Q: Suppose that : R → S is a ring isomorphism. Then g-l is a ring homo- morphism, and hence a ring…

A: According to the given information:

Q: Let R be a ring with m elements. Show that the characteristic of Rdivides m.

A:

Q: Let R be a commutative ring with identity and let I be a proper ideal of R. Prove that R/I is a…

A:

Q: 7. The number of distinct homomorphisms from the ring (Z, +,.) onto the ring (Ze, +,.) is (a) 0 (b)…

A: Number of distinct onto homomorphism is

Q: Let R be a ring with unity 1. If 1 has infinite order under addition then the characteristic of R is…

A: The order of an element a in a ring R is the least positive integer n such that n·a=0. If no such…

Q: Suppose R is a commutative ring and |R|= 30. If I is an ideal of R and |I| = 10, prove that I is a…

A:

Q: Give an example (and justify your choice) of such a subring of a ring P[x] that contains P and is…

A: Given: Ring P[x] and a subring which contains P.

Q: Find all possible ring homomorphisms from phi: Z[i] to Z[i]

A:

Q: Let R be a ring with unity 1, and S = {n.1 : n E Z} . Then S'is Ra subring of Rnot a subring of

A: Let, x ,y in S. So, x = n•1 and, y = m•1 for some n, m in Z.

Q: a. Let R and S be commutative rings with unities and f: R → S be an epimorphism of rings. Prove that…

A: a) Let R and S be commutative rings with unities and f:R→S be epimorphism of rings. Let 0S and 0R…

Q: Let R be a ring with unity 1, and S = {n.1 : n E Z}.Then S'is Ra subring of Rnot a subring of

A: A non-empty set R with two binary operations addition(+) and multiplication(·) is called ring if it…

Q: Let be a commutative ring with unity of characteristic 3. Compute and simplify (a + b)6 for all…

A:

Q: 1. Let R be a commutative ring with unity and let a e Rbe fixed. Prove that the subset Ia = {x E R:…

A: i have provided the detailed proof in next step

Q: The ring Z,3 has exactly-------------maximal ideals

A:

Q: If f:(R.+) - (R', +') be a ring Homomorphism, and R is integral domain, then R' is integral domain…

A:

Q: Let A, B,C be rings. Let & be a ring homomorphism from A into B and ß be a ring homomorphism from B…

A:

Q: Suppose that R and S are commutative rings with identities. Let Ø be a ring homomorphism from R onto…

A:

Q: Let A = {S : S is a subring of C and e S}, and let R = N S be SEF the intersection of all these…

A: Given that A=S| S is a subring of ℂ and 12∈S and let R=∩S∈AS. (a) Consider the ring M=3k: k∈ℤ.…

Q: Show such a subring of the ring P[x], which contains P and is different from P and is not isomorphic…

A: The given details: The ring is P[x]. To show that the subring of the ring P[x] which contains P and…

Q: Suppose R is a commutative ring with 1R# 0R. Show that if f (x) = ao + a1a + a2a ++a,n" is a unit in…

A:

Q: 1. Suppose that p: R → S is a ring homomorphism and that the image of o is not {0}. If R has a unity…

A:

Q: Suppose that R is a commutative ring with unity and that I is an ideal of R. Prove that the set of…

A: Given: R is a commutative ring with unity and that I is an ideal of R. To prove: The set of all x∈R…

Q: (b) Show that if : R→ S is only assumed to be a ring homomorphism, then it is possible to have a…

A: We can prove the above result using two contradictory examples. Note that these examples are only…

Q: The ring Z-[i] has no proper ideals

A:

Q: Let R be a ring with unity e. Verify that the mapping θ: Z---------- R defined by θ (x) = x • e is a…

A: Let R be a ring with unity e, verify the mapping θ:Z→R defined by θx=x.e is a homomorphism If R and…

Q: The ring Zp2, has exactly-----------maximal ideals

A: