Basic Question about a Ring Homomorphisms

In summary, the conversation discussed a book by A.J. Berrick and M.E. Keating called "An Introduction to Rings and Modules With K-Theory in View". The conversation revolved around a remark made by Berrick and Keating regarding ring homomorphisms from the zero or trivial ring. The relevant text from the book states that the obvious map from the zero ring to R is not a ring homomorphism unless R itself is equal to 0. The conversation then delved into trying to understand this statement and clarifying what Berrick and Keating meant by the obvious map. It was concluded that the zero ring does not have a 1 element, which is necessary for (RH2) to hold, and therefore
  • #1
Math Amateur
Gold Member
MHB
3,998
48
I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K).

I need help to clarify a remark of B&K regarding ring homomorphisms from the zero or trivial ring ...

The relevant text from B&K reads as follows:
?temp_hash=93b8052059b87992d1f7fd12348dde3e.png

?temp_hash=93b8052059b87992d1f7fd12348dde3e.png

In the above text from B&K's book we read ...

"... ... This follows from the observation that the obvious map from the zero ring to ##R## is not a ring homomorphism (unless ##R## itself happens to be ##0##). ... ... "
I do not understand the above statement that the obvious map from the zero ring to ##R## is not a ring homomorphism (unless ##R## itself happens to be ##0##) ... ...What, indeed do B&K mean by the obvious map from the zero ring to ##R## ... ... ?It seems to me that the obvious map is a homomorphism ... ..Consider the rings ##T, R## where ##T## is the zero ring and ##R## is any arbitrary ring ... so ##T = \{ 0 \}## where ##0 = 1## ...

Then to me it seems that the "obvious" map is ##f( 0_T) = 0_R## ... ... which seems to me to be a ring homomorphism ...

... BUT ... this must be wrong ... but why ... ?

Can someone please clarify the above for me ...

Some help will be very much appreciated ...

Peter
 

Attachments

  • B&K -1 - Section 1.1.6 Ring Homomorphisms - PART 1 ... ....png
    B&K -1 - Section 1.1.6 Ring Homomorphisms - PART 1 ... ....png
    17.6 KB · Views: 901
  • B&K -2 - Section 1.1.6 Ring Homomorphisms - PART 2 ... ....png
    B&K -2 - Section 1.1.6 Ring Homomorphisms - PART 2 ... ....png
    27 KB · Views: 823
Physics news on Phys.org
  • #2
Math Amateur said:
I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K).

I need help to clarify a remark of B&K regarding ring homomorphisms from the zero or trivial ring ...

The relevant text from B&K reads as follows:
?temp_hash=93b8052059b87992d1f7fd12348dde3e.png

?temp_hash=93b8052059b87992d1f7fd12348dde3e.png

In the above text from B&K's book we read ...

"... ... This follows from the observation that the obvious map from the zero ring to ##R## is not a ring homomorphism (unless ##R## itself happens to be ##0##). ... ... "
I do not understand the above statement that the obvious map from the zero ring to ##R## is not a ring homomorphism (unless ##R## itself happens to be ##0##) ... ...
If you require (RH2) to hold, then you need a ##1_0## which ##\{0\}## doesn't have.
What, indeed do B&K mean by the obvious map from the zero ring to ##R## ... ... ?It seems to me that the obvious map is a homomorphism ... ..Consider the rings ##T, R## where ##T## is the zero ring and ##R## is any arbitrary ring ... so ##T = \{ 0 \}## where ##0 = 1## ...

Then to me it seems that the "obvious" map is ##f( 0_T) = 0_R## ... ... which seems to me to be a ring homomorphism ...
You cannot say ##0=1##. There is no ##1## in ##T=\{0\}##. But your definition of ##f : T \rightarrow R## is correct.
... BUT ... this must be wrong ... but why ... ?
The only wrong here is ##1 \notin \{0\}## and thus (RH2) is violated. (In strict logic, the definition doesn't apply to ##\{0\}## at all, because it is not a ring with unity.)

What can be done with the usage of (RH1) is the following:
##f(r') = f(1_R \cdot r') = f(1_R) \cdot f(r')## so ##f(1_R) = 1_{f(R)}##.

Now this is the crucial point. We cannot conclude ##1_{f(R)} = 1_S## as we do in fields and groups.

E.g. in ##\mathbb{Z_9}## the ring of remainders of division by nine, there is ##1\,\cdot\,3 = 3 = 4 \,\cdot\, 3## but ##1 \neq 4##.
Therefore we need the additional property (RH2) to guarantee ##f(1) = 1##.
 
  • Like
Likes Math Amateur
  • #3
fresh_42 said:
If you require (RH2) to hold, then you need a ##1_0## which ##\{0\}## doesn't have.

It does have one. The zero ring ##\{0\}## has ##0## as ##1_0##, hence ##1=0##. In the definition of a ring with unit, there is usually never the requirement that ##1\neq 0##. That requirement usually is present in fields though.
 
  • Like
Likes Math Amateur
  • #4
micromass said:
It does have one. The zero ring ##\{0\}## has ##0## as ##1_0##, hence ##1=0##. In the definition of a ring with unit, there is usually never the requirement that ##1\neq 0##. That requirement usually is present in fields though.
Thanks. I'm still thinking of an example of a mapping ##f : R \rightarrow S## with ##f(r + s) =f(r)+f(s)## and ##f(rs)=f(r)f(s)## but ##f(1) \neq 1## which is not as simple as ##R=\{0\}##.
 
  • #5
##f:\mathbb{R}\rightarrow \mathbb{R}[X]/(X^2-X):\alpha\rightarrow \alpha X##
 
  • Like
Likes Math Amateur and fresh_42
  • #6
micromass said:
##f:\mathbb{R}\rightarrow \mathbb{R}[X]/(X^2-X):\alpha\rightarrow \alpha X##
Yes. ##r \rightarrow \alpha r## came to my mind, but to get rid of the square this way didn't come to my mind. Silly me, now I feel riled.
 
  • Like
Likes Math Amateur
  • #7
Interesting ring R[X]/(X2 - X). Let's see — if there is a term [c Xn] (c ∈ R, n ≥ 2) in R[X]/(X2 - X) then we can add

[0] ≡ [-c Xn + c Xn-1] ≡ [-c (Xn - Xn-1)] ≡ [-cXn-2(X2 - X)] ≡ [-cXn-2]⋅[X2 - X]​

without changing it, to get (the equivalence class of) a monomial of lower degree. Since any element of R[X] contains only finitely many powers of X in the first place, we can reduce any term to a linear polynomial aX + b with a finite number of such operations.

So the elements of R[X]/(X2 - X) may be thought of by those coefficients (a, b), i.e., as elements of the plane R2. It looks provable that any element of R[X]/(X2 - X) has a unique representation in this form. Then addition is just what you might expect, i.e.,

(a, b) + (c, d) = (a+c, b+d).​

But what is the multiplication rule? I get that (using this representation):

(a, b) ⋅ (c, d) = (ad + bc + ac, bd),
which may be thought of as a map

mult: R2R2.​

Cool!

Question: Is it just a coincidence that (in this representation)

the sum of (the coefficients of the product) = (ad + bc + ac) + bd​

equals

the product of (the sums of coefficients) = (a + b) ⋅ (c + d) ?​
 
Last edited:
  • #8
Thanks to fresh_42, micromass and zinq for the guidance and help ...

BUT ... I am still puzzling over my original question ...

Can you help with the original issue ...

Thanks again for your posts so far ...

Peter
 
  • #9
-----
In the above text from B&K's book we read ...

"... ... This follows from the observation that the obvious map from the zero ring to R is not a ring homomorphism (unless R itself happens to be 0). ... ... "

I do not understand the above statement that the obvious map from the zero ring to R is not a ring homomorphism (unless R itself happens to be 0)
-----

Hint: There are two parts to the book's definition of a ring homomorphism. What happens when you try to prove each one of them to a map from the zero ring to an arbitrary ring? (I did not know where this went wrong until I tried it.) Remember that in the zero ring 0 = 1.
 
  • #10
Math Amateur said:
Thanks to fresh_42, micromass and zinq for the guidance and help ...

BUT ... I am still puzzling over my original question ...

Can you help with the original issue ...

Thanks again for your posts so far ...

Peter
In short: You cannot conclude ##f(1_R) = 1_S## by only the condition (RH1) given.

Homomorphisms of any kind means basically: preserving the structure. Here we have rings.
Thus ##f(r_1+r_2) = f(r_1)+f(r_2)## and ##f(r_1\,\cdot\, r_2) = f(r_1)\,\cdot\,f(r_2)## has to hold.

But what to do with the two neutral elements?
The ##0_R## is no problem, since ##f(0_R)=0_S## already follows from the above.
However, this can't be said about ##1_R##.
But to preserve / respect the whole structure from one ring ##R## to the other ring ##S##, one would expect to have ##f(1_R) = 1_S## as well. As it cannot be proven by (RH1), it is required for ##f## being a ring homomorphism: (RH2).

The examples above (I think all, but I didn't check) map ##1_R \longmapsto 0_S##.
This is something we would not like to be called a ring homomorphism as soon as ##S## has a ##1_S##, too.

But remember post #3. Therefore ##T= \{0\}## counts as an example, where ##1_T = 0_T##. In this case, ##f(1_R)=1_T=0_T## is allowed. There is a unit in both rings and (RH1) and (RH2) hold, even if it happens that in one ring the neutral elements coincide.

Edit: Homomorphism = homo (equal) morph (structure)
That's why, e.g. linear maps - ##L(\alpha x + \beta y) = \alpha L(x) + \beta L(y)## - are also called homomorphisms, or monomorphisms (injective), or epimorphisms (surjective), or isomorphisms (bijective), or automorphisms (bijective within the same underlying set). The same is true for groups, fields, algebras and rings and modules, although they all carry a different structure and therefore a different definition. But the structure itself is preserved by morphisms (name for it if categories are involved).
 
Last edited:
  • #11
one suggestion i have is to just forget about the zero ring as it essentially never arises in practice. At least I have spent my whole 40 year career without encountering it significantly. i.e. in my experience, when it arises it is uninteresting and can be ignored. actual math is usually not about such trivial things. at most you might be troubled now and then to check that your ring you are dealing with is not the zero ring, e.g. that the ideal you are dealing with in a quotient is not the whole ring. i.e. there is a reason it is called the trivial ring.

But this somewhat frivolous comment is just a late night throwaway. If you are enjoying solving the logical puzzle here, go ahead, it may be fun, and even enlightening. But don't lose a lot of sleep over it. It would be more useful to learn the Hilbert nullstellensatz, i.e. a criterion for when an ideal is the unit ideal, (so that if so, then the quotient is the zero ring).
 
Last edited:

Related to Basic Question about a Ring Homomorphisms

1. What is a ring homomorphism?

A ring homomorphism is a mathematical function that maps elements from one ring to another ring while preserving the ring structure. This means that the function preserves the operations of addition and multiplication between elements of the two rings.

2. How is a ring homomorphism different from a group homomorphism?

A ring homomorphism preserves both addition and multiplication, while a group homomorphism only preserves the operation of the group. This means that a ring homomorphism must also preserve the ring's identity element and the distributive property.

3. What is the kernel of a ring homomorphism?

The kernel of a ring homomorphism is the set of elements in the domain ring that are mapped to the additive identity element in the codomain ring. In other words, it is the set of elements that are mapped to zero.

4. Can a ring homomorphism be bijective?

Yes, a ring homomorphism can be bijective, meaning it is both injective (one-to-one) and surjective (onto). This means that for every element in the codomain ring, there is exactly one element in the domain ring that maps to it.

5. Can a ring homomorphism be defined between rings with different operations?

No, a ring homomorphism can only be defined between rings with the same operations of addition and multiplication. This is because a ring homomorphism must preserve these operations, so they must be the same in both rings.

Similar threads

  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Linear and Abstract Algebra
2
Replies
55
Views
4K
Replies
2
Views
2K
  • Linear and Abstract Algebra
Replies
3
Views
2K
  • Linear and Abstract Algebra
Replies
1
Views
2K
  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
6
Views
2K
  • Linear and Abstract Algebra
Replies
4
Views
2K
  • Linear and Abstract Algebra
Replies
7
Views
1K
  • Linear and Abstract Algebra
Replies
2
Views
1K
Back
Top