No Zero Divisors Allowed

Abstract Algebra · Axiom Academy

INTRO No Zero Divisors Allowed Explore multiplication tables to discover which rings have the special property that non-zero times non-zero always gives non-zero. In familiar number systems like the integers, something special happens: if you multiply two non-zero numbers, you always get a non-zero result. Try it! Click any cell in the table to see the multiplication! Let's explore ℤ₅ (integers modulo 5), where we only have elements 0, 1, 2, 3, 4 and arithmetic "wraps around" at 5. Click cells to explore multiplication in ℤ₅ Now let's look at ℤ₆. Will it behave the same way? Explore different rings ℤₙ and discover which have zero divisors. Notice the connection to whether n is prime or composite! In a ring R, a non-zero element a is called a zero divisor if there exists a non-zero element b such that a · b = 0. A commutative ring with unity is called an integral domain (or just domain ) if it has no zero divisors. That is, if a · b = 0, then either a = 0 or b = 0.

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