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2. Domain (ring theory) - Wikipedia

   en.wikipedia.org/wiki/Domain_(ring_theory)

   In algebra, a domain is a nonzero ring in which ab = 0 implies a = 0 or b = 0. [1] (Sometimes such a ring is said to "have the zero-product property".) Equivalently, a domain is a ring in which 0 is the only left zero divisor (or equivalently, the only right zero divisor). A commutative domain is called an integral domain.

3. Linear equation over a ring - Wikipedia

   en.wikipedia.org/wiki/Linear_equation_over_a_ring

   Let R be an effective commutative ring.. There is an algorithm for testing if an element a is a zero divisor: this amounts to solving the linear equation ax = 0.; There is an algorithm for testing if an element a is a unit, and if it is, computing its inverse: this amounts to solving the linear equation ax = 1.

4. Unique factorization domain - Wikipedia

   en.wikipedia.org/wiki/Unique_factorization_domain

   A is a pre-Schreier domain and atomic. A has a divisor theory in which every divisor is principal. A is a Krull domain in which every divisorial ideal is principal (in fact, this is the definition of UFD in Bourbaki.) A is a Krull domain and every prime ideal of height 1 is principal. [7] In practice, (2) and (3) are the most useful conditions ...

5. Exercise (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Exercise_(mathematics)

   A mathematical exercise is a routine application of algebra or other mathematics to a stated challenge. Mathematics teachers assign mathematical exercises to develop the skills of their students. Early exercises deal with addition , subtraction , multiplication , and division of integers .

6. Solving quadratic equations with continued fractions - Wikipedia

   en.wikipedia.org/wiki/Solving_quadratic...

   Notice also that the set obtained by forming all the combinations a + b √ 2, where a and b are integers, is an example of an object known in abstract algebra as a ring, and more specifically as an integral domain. The number ω is a unit in that integral domain. See also algebraic number field.

7. System of linear equations - Wikipedia

   en.wikipedia.org/wiki/System_of_linear_equations

   When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set. For linear equations, logical independence is the same as linear independence. The equations x − 2y = −1, 3x + 5y = 8, and 4x + 3y = 7 are linearly dependent. For example ...