Is R/N a Ring with Unity?

In summary, we are asked to show that if $R$ is a ring with unity and $N$ is an ideal of $R$ such that $N \neq R$, then $R/N$ is a ring with unity. This can be proven by considering the homomorphism $\phi: R \to R/N$ and showing that $1+N$ is the unity of $R/N$. This can be done by noting that for any $r \in R$, $\phi(r) = r + N = \phi(1 \cdot r) = \phi(r \cdot 1) = (1+N)(r+N) = (r+N)(1+N)$. This also assumes the usual operations for factor
  • #1
Fantini
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Good afternoon. Here is the problem:

Show that if $R$ is a ring with unity and $N$ is an ideal of $R$ such that $N \neq R$, then $R/N$ is a ring with unity.

My answer: Consider the homomorphism $\phi: R \to R/N$. Given $r \in R$ we have that $\phi(r) = r + N = \phi(1 \cdot r) = \phi(r \cdot 1) = (1+N)(r+N) = (r+N)(1+N)$, therefore $1+N$ is the unity of $R/N$.

I appreciate the help. Cheers! (Yes)

P.S.: I am assuming the usual operations concerning factor rings:

$(a+N) + (b+N) = (a+b) + N$ and $(a+N)(b+N) = (ab) + N$.
 
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  • #2
Fantini said:
Good afternoon. Here is the problem:

Show that if $R$ is a ring with unity and $N$ is an ideal of $R$ such that $N \neq R$, then $R/N$ is a ring with unity.

My answer: Consider the homomorphism $\phi: R \to R/N$. Given $r \in R$ we have that $\phi(r) = r + N = \phi(1 \cdot r) = \phi(r \cdot 1) = (1+N)(r+N) = (r+N)(1+N)$, therefore $1+N$ is the unity of $R/N$.

I appreciate the help. Cheers! (Yes)

P.S.: I am assuming the usual operations concerning factor rings:

$(a+N) + (b+N) = (a+b) + N$ and $(a+N)(b+N) = (ab) + N$.
Looks right to me. Only thing, there wasn't any need to invoke a homomorphism here. You could just simply show that $(1+N)(r+N)=(r+N)(1+N)=r+N$ for all r in R. Well, there ain't much to show though.
 
  • #3
the only tricky part is that 1+N might actually = N, but this means that 1 is in N,

in which case we still have a multiplicative identity, but:

R/N = R/R = {0}, which is a "silly" ring. this is why we insist N ≠ R at the outset.
 

1. What is a quotient ring with unity?

A quotient ring with unity is a mathematical structure that is created by dividing a ring with unity by an ideal. It is denoted as R/I, where R is the original ring and I is the ideal. The elements of the quotient ring are the cosets of I in R, and the operations on these elements are defined based on the operations in the original ring. The unity of the quotient ring is the coset of the unity of the original ring.

2. How is a quotient ring with unity different from a regular quotient ring?

A regular quotient ring is created by dividing a ring by an ideal. However, in a quotient ring with unity, the ring being divided must already have a unity element. This means that the unity of the original ring must be preserved in the quotient ring. Additionally, the unity of the quotient ring is defined as the coset of the unity of the original ring.

3. What are some applications of quotient rings with unity?

Quotient rings with unity have many applications in abstract algebra, number theory, and algebraic geometry. They are used to study the structure of rings and to solve equations in commutative rings. They are also used in the construction of finite fields, which have applications in coding theory and cryptography.

4. Can a quotient ring with unity have a unity element that is not in the original ring?

No, the unity element of a quotient ring with unity must be a coset of the unity element of the original ring. This is because the unity element must satisfy the same properties in the quotient ring as it does in the original ring. Therefore, it cannot be an element that is not in the original ring.

5. How is the unity element of a quotient ring with unity related to the ideal being divided?

The unity element of a quotient ring with unity is related to the ideal being divided in that it is the coset of the unity element of the original ring in the quotient ring. This means that the ideal being divided, I, is contained in the coset of the unity element. Additionally, the unity element is a multiple of the unity element of I, which is the element that generates the ideal.

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