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In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, =, = = This property is also known as the rule of zero product, the null factor law, the multiplication property of zero, the nonexistence of nontrivial zero divisors, or one of the two zero-factor properties. [1]
The multiplicative identity is 1; anything multiplied by 1 is itself. This feature of 1 is known as the identity property: [26] [27] =. Property of 0 Any number multiplied by 0 is 0. This is known as the zero property of multiplication: [26] = Negation
If the additive identity and the multiplicative identity are the same, then the ring is trivial (proved below). In the ring M m × n (R) of m-by-n matrices over a ring R, the additive identity is the zero matrix, [1] denoted O or 0, and is the m-by-n matrix whose entries consist entirely of the identity element 0 in R.
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors.It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question), just as the empty sum—the result of adding no numbers—is by convention zero, or the additive identity.
Zero is thus an absorbing element. The zero of any ring is also an absorbing element. For an element r of a ring R, r0 = r(0 + 0) = r0 + r0, so 0 = r0, as zero is the unique element a for which r − r = a for any r in the ring R. This property holds true also in a rng since multiplicative identity isn't required.
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The multiplicative identity is often called unity in the latter context (a ring with unity). [10] [11] [12] This should not be confused with a unit in ring theory, which is any element having a multiplicative inverse. By its own definition, unity itself is necessarily a unit. [13] [14]
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