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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.
In a vector space, the additive inverse −v (often called the opposite vector of v) has the same magnitude as v and but the opposite direction. [11] In modular arithmetic, the modular additive inverse of x is the number a such that a + x ≡ 0 (mod n) and always exists. For example, the inverse of 3 modulo 11 is 8, as 3 + 8 ≡ 0 (mod 11). [12]
The axioms of modules imply that (−1)x = −x, where the first minus denotes the additive inverse in the ring and the second minus the additive inverse in the module. Using this and denoting repeated addition by a multiplication by a positive integer allows identifying abelian groups with modules over the ring of integers.
The New York Times. ... Here are additional clues for each of the words in today's Mini Crossword. NYT Mini Across Hints. 1 Across: Chief Norse god — HINT: It starts with the letter "O"
Answers to NYT's The Mini Crossword for Saturday, January 18, 2025 Don't go any further unless you want to know exactly what the correct words are in today's Mini Crossword. NYT Mini Across Answers
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.
Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse. At the same time, semirings are a generalization of bounded distributive lattices. The smallest semiring that is not a ring is the two-element Boolean algebra, for instance with logical disjunction as
A ring in which all elements are idempotent is called a Boolean ring. Some authors use the term "idempotent ring" for this type of ring. In such a ring, multiplication is commutative and every element is its own additive inverse. A ring is semisimple if and only if every right (or every left) ideal is generated by an idempotent.