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If a × b = a × c, then it does not follow that b = c even if a ≠ 0 (take c = b + a for example) Matrix multiplication also does not necessarily obey the cancellation law. If AB = AC and A ≠ 0, then one must show that matrix A is invertible (i.e. has det(A) ≠ 0) before one can conclude that B = C. If det(A) = 0, then B might not equal C ...
In mathematics, a cancellative semigroup (also called a cancellation semigroup) is a semigroup having the cancellation property. [1] In intuitive terms, the cancellation property asserts that from an equality of the form a·b = a·c, where · is a binary operation, one can cancel the element a and deduce the equality b = c.
A monoid (M, •) has the cancellation property (or is cancellative) if for all a, b and c in M, the equality a • b = a • c implies b = c, and the equality b • a = c • a implies b = c. A commutative monoid with the cancellation property can always be embedded in a group via the Grothendieck group construction.
[1] [2] Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c. "Integral domain" is defined almost universally as above, but there is some variation.
If an AF algebra A = (∪ n A n) −, then an ideal J in A takes the form ∪ n (J ∩ A n) −. In particular, J is itself an AF algebra. Given a Bratteli diagram of A and some subset S of nodes, the subdiagram generated by S gives inductive system that specifies an ideal of A. In fact, every ideal arises in this way.
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Let (∗) be a C′(1/6) presentation. Then an element g in G has order n > 1 if and only if there is a relator r in R of the form r = s n in F(X) such that g is conjugate to s in G. In particular, if all elements of R are not proper powers in F(X) then G is torsion-free. If (∗) is a finite C′(1/6) presentation, the group G is word-hyperbolic.
Key takeaways. You will receive a 1099-C Cancellation of Debt form if a lender forgives more than $600 of taxable debt. You must include the amount of canceled debt on your federal tax return as a ...