Zero Divisors:
(0,0), (0,1), (0,2), (0,3), (1,0), (1,2)
Units:
(1,1), (1,3)
(1,1)*(1,1) = (1,1), so (1,1)^{-1} = (1,1)
(1,3)*(1,3) = (1,1), so (1,3)^{-1} = (1,3)
Neither:
None
Z2 × Z4 is a ring under component-wise addition and multiplication:
(a, b) + (c, d) = (a + c, b + d)
and
(a, b) · (c, d) = (ac, bd).
Classify each element of Z2 × Z4 as a zero divisor, a unit, or neither.
If the element is a zero divisor, ﬁnd a nonzero element whose product with the ﬁrst element is (0,0).
If the element is a unit, ﬁnd its multiplicative inverse.
My Answer:
Z2 x Z4 = { (0,0), (0,1), (0,2), (0,3), (1,0), (1,1), (1,2), (1,3) }
Zero Divisors:
(0,0)*(0,0) = (0,0)
(0,1)*(1,0) = (0,0)
(0,2)*(0,2) = (0,0)
(0,3)*(1,0) = (0,0)
(1,0)*(1,0) = (0,0)
(1,2)*(0,2) = (0,0)
Unit:
(1,1)
Neither:
(0,3)
I didn’t see the method by looking at the answer. So I made the addition and multiplication tables, and then I saw the method (ab is shortcut for (a,b)) :
Divisors of 0 (ab=0): What do you multiply by to get 00
00 01 02 03 10 11 12 13
00 10 12 10 01 .... 02 ..
Additive inverse (a+b=0) : What do you add to get 00
00 01 02 03 10 11 12 13
00 03 02 01 10 13 12 13
Units (ab=1): What do you multiply by to get 11
00 01 02 03 10 11 12 13
......................11..... 13
Definition:
1) amodn is remainder of a/n
5mod4=1
2) (a+b)=(a+b)modn, (a*b)=(a*b)modn
2+2=4mod4=0, 3*3=9mod4=1
3) Zn: 0,1,2,3,…n with above rules for addition and multiplication.
For the present example, each component satisfies the Definitions:
(1,3)+(1,3)=(2mod2,6mod4)=(0,2)
(1,3)*(1,3)=(1mod2,9mod4)=(1,1)
One can always try trial-and-error, but that is not the most efficient approach, usually.
Let's break the elements of $\Bbb Z_2 \times \Bbb Z_4$ into 4 types:
1: (0,0)
2: (1,0)
3: (0,a), a = 1,2,3
4: (1,a), a = 1,2,3
It should be obvious that types 1-3 are all zero divisors. The zero of a ring is (obviously) always a zero-divisor. For those elements where 0 is one of the coordinates, multiplying by an element where 0 is in the OTHER coordinate will always gives you 0, and we can always pick the SAME coordinate in the second element to be non-zero, that is:
type 2 times type 3 = (0,0), so these are all zero divisors.
That leaves 3 elements to check: (1,1),(1,2), and (1,3). (1,1) is a unit, being the ring identity (which is always a unit), so we only need to look at (1,2) and (1,3).
An observation:
(1,2)(a,b) = (a,2b). If (1,2) were a unit, we would have to have: a = 1 (mod 2),2b = 1 (mod 4). Similarly, if (1,2) were a zero divisor, we would have to have a = 0 (mod 2), 2b = 0 (mod 4), and since (0,b) cannot be (0,0), we need a non-zero such b.
Similar considerations hold for (1,3), which is a unit iff there exists b with 3b = 1 (mod 4), and is a zero divisor iff there exists non-zero b with 3b = 0 (mod 4).
In short, we have to look at the units and zero-divisors of $\Bbb Z_4$.
Well, in $\Bbb Z_4$ we have:
2*2 = 0 (mod 4), so 2 is a zero-divisor. this means that (1,2) is likewise a zero-divisor, explicitly: (1,2)*(0,2) = (0,0), and (0,2) is not (0,0).
In $\Bbb Z_4$, 3 is a unit, since 3*3 = 1 (mod 4). this means that (1,3) is likewise a unit, explicitly: (1,3)*(1,3) = (1,1).
IN GENERAL, for $\Bbb Z_m \times \Bbb Z_n$, we have:
(0,b) and (a,0) are zero divisors for ANY a,b.
(a,b) is a zero divisor if either a OR b is a zero-divisor (or both).
(a,b) is a unit if BOTH a and b are units in their respective rings.
********
It is possible for elements of a ring $R \times S$ to be neither, although that does not happen if $R,S$ are finite cyclic rings. An example is given by $R = \Bbb Z, S = \Bbb Z_2$, where the element $(2,1)$ is neither:
(2,1)*(a,b) = (0,0) implies (2a,b) = (0,0), which implies b = 0, and 2a = 0. The only integer a for which 2a = 0, is 0, so (2,1) is not a zero-divisor.
(2,1)*(a,b) = (1,1) implies (2a,b) = (1,1), which implies b = 1, and 2a = 1. There is NO integer a for which 2a = 1 (1/2 is NOT an integer), so (2,1) is not a unit.
Deveno, why is your description shorter or clearer than simply writing out the elements of (a,b) and underneath, by inspection, respectively listing the divisors of 0, additive inverse, and units. That is not trial and error, it is a clear, simple, systematic, way to do it. I would think you don’t have to belabor the fact that, for example, if you have (0,b) or (a,0), there is nothing you can multiply 0 by to get 1.
Deveno uses the qualifier "usually" when stating that writing out every element (a,b) is not the most efficient approach. In this case, the ring is small enough that it is not terribly difficult to write out a complete multiplication table. However, if the ring were , you would have a ring with elements. Deveno's method of computing the zero divisors and units still works while trying to write out the possible multiplications of the ring would take quite a long time.
So, Deveno attempted to give the OP some theory that could help understand future problems that do not use such small groups. The idea being that this knowledge could be useful in solving more complex problems of a similar nature. However, in this example, I agree with you that either method works. Still, there is no harm in imparting a little extra knowledge, is there?
The first part of Deveno’s post is simply a belaboring of the obvious, it was done in post #3 by inspection.
Of the three rules given at the end of Deveno’s post, the first two are wrong, and there is no ”theory.”
By Definition (as given in my post #3):
1) (a,b) is a divisor of 0 if a and b are divisors of 0.
2) (a,b) is a unit if a and b are units.
Perhaps what Deveno wrote is not connecting with you, but it may be helpful to others. Ring theory is "theory" so I don't understand why you dislike my use of that word. And I don't understand why you say the first two rules Deveno posted are "wrong". For example, you showed that (1,2) is a zero divisor, but 1 is not a zero divisor in . So, your "rule 1" seems inaccurate.
The OP originally thought one of the elements of the ring was neither a zero-divisor nor a unit. Deveno brought up to discuss such elements by pointing out that every element of any finite product of finite cyclic rings is either a zero-divisor or a unit. Then, to expand upon that, he gave an example of a ring containing an element that is neither a zero-divisor nor a unit. Exposure to concrete examples is frequently useful to students learning ring theory for the first time.
By your (1), in , is not a zero-divisor since 1 is not a zero-divisor in . However, in post #3, you show the multiplication that proves that (1,0) is a zero divisor. Therefore, (1) is not correct.
That was Deveno's point exactly. I'm glad you agree.
What is (Z/2Z)X(Z/4Z) and what does it have to do with Z2XZ4?
(1,0) is not a divisor of zero in Z2/Z4. I made a mistake, as can easily be seen from defintion 1) below. 1 is not a zero of Z2. Without all the unintelligible, undefined blah blah.
Deveno's point was:
In ZXZ2:
"(2,1)*(a,b) = (0,0) implies (2a,b) = (0,0), which implies b = 0, and 2a = 0. The only integer a for which 2a = 0, is 0, so (2,1) is not a zero-divisor.
(2,1)*(a,b) = (1,1) implies (2a,b) = (1,1), which implies b = 1, and 2a = 1. There is NO integer a for which 2a = 1 (1/2 is NOT an integer), so (2,1) is not a unit.
My point was:
(2,1) is neither a unit or divisor of ZXZ2 by definiton. (2 is neither a unit or divisor of 0 in Z)
By Definition (as given in my post #3):
1) (a,b) is a divisor of 0 if a and b are divisors of 0.
2) (a,b) is a unit if a and b are units.
My point is general and simple. If you don't see the difference, sorry.
is always the cyclic ring of order 2. is the notation for the ring of 2-adic integers. I avoid using to avoid confusion. So, if you don't like my notation, that is your preference, and and .
Your definition (as given in post #3) is a different definition from the standard definition of a zero divisor of a ring. The standard definition (Wikipedia - Zero divisor):
As you showed in post 3, . Since , this satisfies the definition so is a zero-divisor (even though 1 is not a zero-divisor).In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero such that , or equivalently if the map from to that sends to is not injective.
So, I don't know where you got your definitions (as given in your post #3), but they are not standard.
MY SUMMARY:
Post 2 answered OP correctly without any explanation.
Post 3 gave a systematic method for answering OP with general defs of zero divisors and units: ab=0, and ab=1. Missing: a≠0 and b≠0 for ab=0. (but very limited)
Post 4:
1) (0,b) and (a,0) are zero divisors for ANY a,b.
2) (a,b) is a zero divisor if either a OR b is a zero-divisor (or both).
3) (a,b) is a unit if BOTH a and b are units in their respective rings.
Comment:
1) a and b both ≠ 0
2) if a is a zero divisor: ac=0, a≠0, c≠0. Then
(a,b)(c,0)=(a,0)(c,b)=(0,0), any b.
CONCLUSION:
Post 4 answered the OP, which I realized after I got past the cramped wordiness, which is a put-off.
Turned out to be an interesting, educational thread, except for the irrelevant obtuse abstract jargon copied from wiki articles.