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Topic: Finite Rings
Replies: 28   Last Post: Feb 6, 2013 8:33 AM

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 magidin@math.berkeley.edu Posts: 11,697 Registered: 12/4/04
Re: Finite Rings
Posted: Feb 4, 2013 11:55 AM

On Monday, February 4, 2013 2:26:16 AM UTC-6, William Elliot wrote:
> On Sun, 3 Feb 2013, Arturo Magidin wrote:
>

> > On Sunday, February 3, 2013 10:46:14 PM UTC-6, William Elliot wrote:
>
> > > On Sun, 3 Feb 2013, Arturo Magidin wrote:
>
> > > > On Sunday, February 3, 2013 9:21:19 PM UTC-6, William Elliot wrote:
>
>
>

> > > > > > If R is a finite commutative ring without multiplicative identity
>
> > > > > > and if every element is a zero divisor, then does there exist
>
> > > > > > a nonzero element which annihilates all elements of the ring?
>
>
>

> > Which means that the trivial ring DOES NOT satisfy the hypothesis, and
>
> > therefore is not to be considered, period. The fact that a ring without
>
> > multiplicative identity must contain a nonzero element need not be a
>
> > premise, because "does not have a multiplicative identity" IMPLIES,
>
> > necessarily, the existence of a nonzero identity.
>
>
>
> Clever but no better.

Nothing clever about it. Just following the standard definitions.

> According to John Beachy in "Abstract Algebra,"
>
> the multiplicative identity is distinct from the additive identity.

Here's a summary from my bookshelf.

--
Bourbaki, "Algebra", I, Section 8, paragraph 1.

Definition 1. A ring is a set A with two laws of composition called respectively addition and multiplication, satisfying the following axioms:

(AN I) Under addition A is a commutative group.
(AN II) Multiplication is associative and possesses an identity element.
(AN III) Multiplication is distributive with respect to addition.
The ring A is said to be commutative if its multiplication is commutative.
--

No requirement that 1=/=0.

--
Dummit and Foote, "Abstract Algebra, 2nd Edition"; Chapter 7, section 7.1

Definition. (1) A ring R is a set together with two binary operations + and x (called addition and multiplication) satisfying the following axioms:

(i) (R,+) is an abelian group.
(ii) x is associative: (a x b) x c = a x (b x c) for all a,b,c in R,
(iii) The distributive law holds in R: for all a,b,c in R,
(a + b) x c = (a x c) + (b x c) and a x (b + c) = (a x b) + (a x c).
(2) The ring R is commutative if multiplication is commutative.
(3) The ring R is said to have an identity (or contain a 1) if there exists an element 1 in R with 1 x a = a x 1 = a for all a in R.

--
Again, no requirement that 1 =/= 0.

--
Eisenbud, "Commutative Algebra with a view towards Algebraic Geometry", Chapter 0, section 1.

A ring is an abelian group R with a multiplication operation (a,b) |-> ab and an identity element 1, satisfying, for all a,b,c in R:

a(bc) = (ab)c (associativity)
a(b+c) = ab + ac
(b+c)a = ba + ca (distributivity)
1a = a1 = a (identity)

--
Again, no requirement that 1=/=0.

--

Herstein's "Topics in Algebra" (translating from the Spanish edition by Trillas), Chapter 3, section 1.

Definition. A nonempty set R is called an associative ring if R has two operations, denoted by + and x, respectively, such that for all a,b,c in R

(1) a+b is in R
(2) a+b = b+a
(3) (a+b)+c = a+(b+c)
(4) There is an element 0 in R such that a+0 = a for all a in R.
(5) There is an element -a in R such that a+(-a)=0.
(6) a x b is in R
(7) a x (b x c) = (a x b) x c
(8) a x (b+c) = (a x b) + (a x c) and (b+c) x a = (b x a) + (c x a) (the two distributive laws).

[...]
It may or may not happen that there is an element 1 in R such that a x 1 = 1 x a = a for all a in R. If such an element exists, we will say that R is a "ring with unit element".

--

Again, no requirement that 1 =/= 0.

--

Hungerford, "Algebra", Chapter III, section 1.

Definition 1.1. A ring is a nonempty set R together with two binary distributive operations (usually denoted as addition (+) and multiplication) such that:

(i) (R,+) is an abelian group;
(ii) (ab)c = a(bc) for all a,b,c in R (associative multiplication)
(iii) a(b+c) = ab + ac and (a+b)c = ac + bc (left and right distributive laws).

(iv) ab = ba for all a,b in R,

then R is said to be a commutitive ring. If R contains an element 1_R such that

(v) 1_R a = a 1_R = a for all a in R

then R is said to be a ring with identity.

--

Again, no requirement that 1=/=0.

--

Lang's "Algebra" (Revised Third Edition), Chapter II, section 1. After the definition,

As usual, we denote the unit element for addition by 0, and the unit element for multiplication by 1. We do not assume that 1=/=0.

--

Explicit statement that the assumption 1=/=0 is not included.

--

Rotman, "Advanced Modern Algebra, 2nd Edition", Chapter 2, Section 2.1.

After the definition, it says:

"The element 1in a ring R has several names; it is called 'one', the 'unit' of R, or the 'identity' of R. We do not assume that 1=/=0, but see Proposition 2.2(ii)."

(Which reads "If 1=0, then R consists of the single element 0. In this case, R is called the 'zero ring'")

--
Again, an explicit inclusion of the case 1=0.

The only two books I have that do not allow 1=0 are: Zariski and Samuel's "Commutative Algebra", which restricts the use of the term "identity" to rings that are not nullrings; and Lam's "A First Course in Noncommutative rings" and "Lectures on Rings and Modules", which specifies this explicitly in the introduction.

> So rather that toss bull about,

You are confused. The bull tosser extraordinaire in this group is you.

> what definitions are you using?

The standard one. As evidenced above.

--
Arturo Magidin