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In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “universal” group having these properties, in the sense that any two homomorphisms from G and H into a group K factor uniquely through a ...
In algebra, the free product ( coproduct) of a family of associative algebras over a commutative ring R is the associative algebra over R that is, roughly, defined by the generators and the relations of the 's. The free product of two algebras A, B is denoted by A ∗ B. The notion is a ring-theoretic analog of a free product of groups .
The direct product of a family of R-algebras is the ring-theoretic direct product. This becomes an R-algebra with the obvious scalar multiplication. Free products One can form a free product of R-algebras in a manner similar to the free product of groups. The free product is the coproduct in the category of R-algebras. Tensor products
In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an ...
The tensor product can be used as a means of taking intersections of two subschemes in a scheme: consider the -algebras , , then their tensor product is , which describes the intersection of the algebraic curves f = 0 and g = 0 in the affine plane over C. More generally, if is a commutative ring and are ideals, then , with a unique isomorphism ...
Product (mathematics) In mathematics, a product is the result of multiplication, or an expression that identifies objects (numbers or variables) to be multiplied, called factors. For example, 21 is the product of 3 and 7 (the result of multiplication), and is the product of and (indicating that the two factors should be multiplied together).
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