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2. Multiplicative inverse - Wikipedia

   en.wikipedia.org/wiki/Multiplicative_inverse

   The sedenions are an algebra in which every nonzero element has a multiplicative inverse, but which nonetheless has divisors of zero, that is, nonzero elements x, y such that xy = 0. A square matrix has an inverse if and only if its determinant has an inverse in the coefficient ring .

3. Inverse element - Wikipedia

   en.wikipedia.org/wiki/Inverse_element

   The inverse or multiplicative inverse (for avoiding confusion with additive inverses) of a unit x is denoted , or, when the multiplication is commutative, . The additive identity 0 is never a unit, except when the ring is the zero ring, which has 0 as its unique element.

4. Ring (mathematics) - Wikipedia

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

   Books on commutative algebra or algebraic geometry often adopt the convention that ring means commutative ring, to simplify terminology. In a ring, multiplicative inverses are not required to exist. A nonzero commutative ring in which every nonzero element has a multiplicative inverse is called a field.

5. Unit (ring theory) - Wikipedia

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

   The multiplicative identity 1 and its additive inverse −1 are always units. More generally, any root of unity in a ring R is a unit: if r n = 1, then r n−1 is a multiplicative inverse of r. In a nonzero ring, the element 0 is not a unit, so R × is not closed under addition.

6. Field (mathematics) - Wikipedia

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

   Informally, a field is a set, along with two operations defined on that set: an addition operation written as a + b, and a multiplication operation written as a ⋅ b, both of which behave similarly as they behave for rational numbers and real numbers, including the existence of an additive inverse −a for all elements a, and of a multiplicative inverse b −1 for every nonzero element b.

7. Kernel (algebra) - Wikipedia

   en.wikipedia.org/wiki/Kernel_(algebra)

   The first isomorphism theorem in general universal algebra states that this quotient algebra is naturally isomorphic to the image of f (which is a subalgebra of B). Note that the definition of kernel here (as in the monoid example) doesn't depend on the algebraic structure; it is a purely set-theoretic concept.

8. Semiring - Wikipedia

   en.wikipedia.org/wiki/Semiring

   This contrast is also why for the theory of semirings, the multiplicative zero must be specified explicitly. Here − 1 {\displaystyle -1} , the additive inverse of 1 {\displaystyle 1} , squares to 1 {\displaystyle 1} .

9. Invertible matrix - Wikipedia

   en.wikipedia.org/wiki/Invertible_matrix

   If this is the case, then the matrix B is uniquely determined by A, and is called the (multiplicative) inverse of A, denoted by A −1. Matrix inversion is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix. [2] Over a field, a square matrix that is not invertible is called singular or ...