Search results
Results from the WOW.Com Content Network
The Buchner ring expansion is a two-step organic C-C bond forming reaction used to access 7-membered rings. The first step involves formation of a carbene from ethyl diazoacetate, which cyclopropanates an aromatic ring. The ring expansion occurs in the second step, with an electrocyclic reaction opening the cyclopropane ring to form the 7 ...
A common method for expanding a ring involves opening cyclopropane-containing bicyclic intermediate. The strategy can start with a Simmons-Smith-like cyclopropanation of a cyclic alkene. [3] A related cyclopropane-based ring expansion is the Buchner ring expansion. The Buchner ring expansion is used to convert arenes to cycloheptatrienes. The ...
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...
Explicitly, if E is a subset of F, then the quotient ring of F by the ideal generated by E is called the ring with generators X and relations E. If we used a ring, say, A as a base ring instead of Z , {\displaystyle \mathbb {Z} ,} then the resulting ring will be over A .
In fact, x ≡ b m n −1 m + a n m −1 n (mod mn) where m n −1 is the inverse of m modulo n and n m −1 is the inverse of n modulo m. Lagrange's theorem: If p is prime and f (x) = a 0 x d + ... + a d is a polynomial with integer coefficients such that p is not a divisor of a 0, then the congruence f (x) ≡ 0 (mod p) has at most d non ...
Given a commutative ring A, an A-extension or an extension of an A-algebra is defined in the same way by replacing "ring" with "algebra over A" and "abelian groups" with "A-modules". An extension is said to be trivial or to split if ϕ {\displaystyle \phi } splits; i.e., ϕ {\displaystyle \phi } admits a section that is a ring homomorphism [ 2 ...
On 'Antiques Roadshow,' a woman brought in a ring she won at an auction about two or three years ago. She paid just $30 for it, thinking it was just a cubic zirconia. When she got it home, she ...
In mathematics, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers The main article for this category is Ring theory .