Abstract Algebra, rings, zero divisors, and cartesian product

In summary, the conversation discusses the Cartesian Product of two nonzero rings, R and S, and how to show that the product contains zero divisors. The definition of multiplication in the Cartesian Product is also mentioned, along with the conditions for r1r2 and s1s2 to equal 0. However, it is determined that neither r1 nor s1 can equal 0, leading to the conclusion that R x S contains zero divisors.
  • #1
jamestrodden
1
0
The problem states:
Let R and S be nonzero rings. Show that R x S contains zero divisors.

I had to look up what a nonzero ring was. This means the ring contains at least one nonzero element.

R x S is the Cartesian Product so if we have two rings R and S
If r1 r2 belong to R and s1 s1 belong to S

(r1, s1) + (r2,s2) = (r1+r2, s1 + s2)
I am using * to denote multiplication here.
(r1, s1)*(r2,s2) = (r1r2,s1s2)

Since we are talking about zero divisors I am going to need the definition of multiplication in the Cartesian Product.

I mean i going to need that (r1r2, s1s2 ) = (0,0)

so r1r2 = 0 and s1s2 = 0 . But neither r1 = 0 = r2 or s1 = 0 = s2
 
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  • #2
jamestrodden said:
so r1r2 = 0 and s1s2 = 0 . But neither r1 = 0 = r2 or s1 = 0 = s2
Why'd you stop here?
 

Related to Abstract Algebra, rings, zero divisors, and cartesian product

1. What is Abstract Algebra?

Abstract Algebra is a branch of mathematics that deals with the study of algebraic structures, such as groups, rings, and fields. It focuses on the abstract properties and structures of these mathematical objects rather than specific numbers or equations.

2. What are rings in Abstract Algebra?

In Abstract Algebra, a ring is a mathematical structure that consists of a set of elements and two binary operations, usually addition and multiplication. It follows certain axioms and properties, such as closure, associativity, and distributivity, which make it a powerful tool in studying abstract algebraic structures.

3. What are zero divisors in Abstract Algebra?

A zero divisor in Abstract Algebra is an element of a ring that, when multiplied by another element, yields a product of zero. In other words, it is an element that has a non-zero product with at least one other element in the ring. Zero divisors play a significant role in determining the structure and properties of a ring.

4. What is the cartesian product in Abstract Algebra?

The cartesian product, also known as the direct product, is an operation in Abstract Algebra that combines two sets to form a new set. In the context of rings, the cartesian product combines two rings to create a new ring with elements that are ordered pairs from the original rings. It is denoted by the symbol ×.

5. How is Abstract Algebra applied in real life?

Abstract Algebra has many practical applications in various fields, such as computer science, cryptography, and physics. For example, it is used in coding theory to create efficient error-correcting codes, in cryptography to secure data transmission, and in quantum mechanics to describe the behavior of particles. It also has applications in data analysis, economics, and engineering.

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