Second Isomorphism Theorem for Rings .... Bland Theorem 3.3.15 .... ....

In summary, Bland's Second Isomorphism Theorem for rings states that the kernel of the canonical ring homomorphism from $R$ to $R/I_2$ is $I_1/I_2$. This can be easily shown using the identity of $R/I_1$ being $I_1$ and the fact that the cosets in the kernel are those that contain elements of $I_1$. It is also important to note that the quotient ring $R/I_1$ is a ring under addition and multiplication of cosets, but the definition of cosets only involves addition.
  • #1
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I am reading "The Basics of Abstract Algebra" by Paul E. Bland ... ...

I am currently focused on Chapter 3: Sets with Two Binary Operations: Rings ... ...

I need help with Bland's proof of the Second Isomorphism Theorem for rings ...

Bland's Second Isomorphism Theorem for rings and its proof read as follows:
View attachment 7969
https://www.physicsforums.com/attachments/7970
In the above proof by Bland we read the following:

" ... ... This map is easily shown to be a well defined ring homomorphism with kernel \(\displaystyle I_1/I_2\). ... ... "I can see that \(\displaystyle f\) is a ring homomorphism ... but how do we prove that the kernel is \(\displaystyle I_1/I_2\) ... ... ?Hope someone can help ...

Peter
 
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  • #2
Hi Peter,

The identity of $R/I_1$ is $I_1$. $\ker f$ contains the cosets $x+I_2$ such that $x+I_1 = I_1$, which means that $x\in I_1$.

This shows that $\ker f$ is the set of cosets $\{(x+I_2)\mid x\in I_1\}$, which is $I_1/I_2$.
 
  • #3
castor28 said:
Hi Peter,

The identity of $R/I_1$ is $I_1$. $\ker f$ contains the cosets $x+I_2$ such that $x+I_1 = I_1$, which means that $x\in I_1$.

This shows that $\ker f$ is the set of cosets $\{(x+I_2)\mid x\in I_1\}$, which is $I_1/I_2$.
Thanks for the help, castor28 ...

But just a point of clarification ...

You write: " ... ... The identity of $R/I_1$ is $I_1$ ... ... "Surely the identity of \(\displaystyle R/I_1\) is the coset \(\displaystyle 1_R + I_1\) because \(\displaystyle (a + I_1) ( 1_R + I_1 ) = a 1_R + I_1 = a + I_1\) ... and similarly \(\displaystyle ( 1_R + I_1 ) ( a + I_1) = a + I_1\) ...

Can you please clarify ...

Peter
 
  • #4
Peter said:
Thanks for the help, castor28 ...

But just a point of clarification ...

You write: " ... ... The identity of $R/I_1$ is $I_1$ ... ... "Surely the identity of \(\displaystyle R/I_1\) is the coset \(\displaystyle 1_R + I_1\) because \(\displaystyle (a + I_1) ( 1_R + I_1 ) = a 1_R + I_1 = a + I_1\) ... and similarly \(\displaystyle ( 1_R + I_1 ) ( a + I_1) = a + I_1\) ...

Can you please clarify ...

Peter
Hi Peter,

Quotient rings are defined in terms of the additive group; that is why cosets of an ideal $I$ are written as $x+I$. The identity of the additive group is $0$, and the identity of $R/I$ is the coset containing $0$, which is obviously $I$. The kernel is defined as the inverse image of the identity of the additive group.

The coset $1+I$ is the multiplicative identity of the quotient ring $R/I$.
 
  • #5
castor28 said:
Hi Peter,

Quotient rings are defined in terms of the additive group; that is why cosets of an ideal $I$ are written as $x+I$. The identity of the additive group is $0$, and the identity of $R/I$ is the coset containing $0$, which is obviously $I$. The kernel is defined as the inverse image of the identity of the additive group.

The coset $1+I$ is the multiplicative identity of the quotient ring $R/I$.

Thanks castor28 ...

Hmm ... beginning to understand what you are saying ...

Still concerned and a bit confused ...

Surely \(\displaystyle R/I\) is a ring under the addition and multiplication of cosets ...
and hence has a (multiplicative) identity as I described ..

But the definition of cosets of course involves only addition ...

Is that correct?

Peter
 
  • #6
Peter said:
Thanks castor28 ...

Hmm ... beginning to understand what you are saying ...

Still concerned and a bit confused ...

Surely \(\displaystyle R/I\) is a ring under the addition and multiplication of cosets ...
and hence has a (multiplicative) identity as I described ..

But the definition of cosets of course involves only addition ...

Is that correct?

Peter
Hi Peter,

Yes, that is correct.

If $I$ is a subgroup (necessarily normal) of the additive group of $R$, the quotient group $R/I$ is an additive group, and there is a canonical group homomorphism $f:R\to R/I$ that sends $x\in R$ to the coset $x+I$. The additive identity of $R/I$ is the coset $0+I=I$. $\ker f$ is the inverse image of that coset.

If we add the stronger condition that $I$ is an ideal of $R$, then it can be shown that multiplication of cosets is well-defined, and $f$ is also a ring homomorphism.

Note, however, that $R$ is not a group under multiplication (unless it is trivial); therefore, the (group-related) concept of kernel cannot be based on the multiplicative structure.
 
  • #7
The well-definedness of the ring-homomorphism is explained to you in this thread:
https://mathhelpboards.com/linear-abstract-algebra-14/quotient-rings-remarks-adkins-weintraub-23836.html?highlight=adkins
 

Related to Second Isomorphism Theorem for Rings .... Bland Theorem 3.3.15 .... ....

1. What is the Second Isomorphism Theorem for Rings?

The Second Isomorphism Theorem for Rings, also known as Bland Theorem 3.3.15, states that if R is a ring, S and T are subrings of R, and S is a normal subring of R, then the quotient ring (R/T)/(S/T) is isomorphic to the quotient ring R/S.

2. What is the significance of the Second Isomorphism Theorem for Rings?

The Second Isomorphism Theorem for Rings allows us to better understand the structure of rings and how they relate to their subrings. It also provides a useful tool for proving other theorems in ring theory.

3. How does the Second Isomorphism Theorem for Rings differ from the First and Third Isomorphism Theorems?

The First Isomorphism Theorem for Rings states that if f: R -> S is a homomorphism of rings, then the kernel of f is a normal subring of R and the image of f is isomorphic to the quotient ring R/ker(f). The Third Isomorphism Theorem states that if I and J are ideals in a ring R, and J is a subset of I, then (R/I)/(J/I) is isomorphic to R/J. The Second Isomorphism Theorem is a combination of these two theorems, with the added condition that S is a normal subring of R.

4. What are some applications of the Second Isomorphism Theorem for Rings?

The Second Isomorphism Theorem for Rings can be applied to various areas of mathematics, including group theory, algebraic geometry, and coding theory. It can also be used in the study of finite rings and fields, as well as in the construction of quotient rings in ring theory.

5. Are there any limitations to the Second Isomorphism Theorem for Rings?

The Second Isomorphism Theorem for Rings is limited to the case where S is a normal subring of R. It also does not hold for non-commutative rings. Additionally, the theorem only applies to rings and does not extend to other algebraic structures such as groups or fields.

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