May 4, 2018 · Why do some sources define zero divisors in such a way that $0$ is a zero divisor? Does it not make things confusing? For example, in this terminology integral domains actually have zero …

A small part answer: It is not so much what they are good for, it is more that they are things to avoid. The familiar algebra can often be extended to the elements of a ring that are not zero divisors.

Dec 28, 2015 · 8 All positive and negative numbers including zero are called integers. So in the form a = bq a = b q, since 0 = 0ㆍq 0 = 0 ㆍ q is true for any integer q q, 0 0 can have 0 0 as a divisor of itself …

Mar 12, 2021 · Hint: is a proper divisor of $n$ that is not equal to $1$ a zero divisor in $\mathbb {Z}/n\mathbb {Z}$? As to claim 2, do you consider $0$ to be a zero divisor? Some ...

Apr 14, 2017 · The definition of a zero divisor is exactly what it says on the tin; recall that normally we say a|b a | b if we can write b = ak b = a k so that the quotient k = b/a ∈Z k = b / a ∈ Z. Let b = 0 b = 0.

Jan 14, 2015 · You seem to have proven that no element can be both a unit and a zero divisor, which is a little different from the statement that everything is either a unit or a zero divisor.

Jul 15, 2015 · Is every noninvertible matrix over a field a zero divisor? Related to this: What are sufficient conditions for a matrix to be a zero divisor over a noncommutative ring?

Thomas is spot on: zero definitely is a zero divisor according to any reasonable definition of the term. Moreover, many well-known theorems would be false if one took the absurd position that zero is not …

In "Basic Algebra" by Robert Ash, p 29, he says "If a and b are nonzero but ab=0, we say that a and b are zero divisors". So again Ash leaves open what to say about 0, as does Definition 2 above. And …

Oct 22, 2021 · @Bliipbluup No, no algorithm. I just thought of a simple non-commutative ring and tried the first non-zero divisors that came to mind.