Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, it is possible to combine several rings into one large product ring. This is done as follows: if I is some index set and Ri is a ring for every i in I, then the cartesian product i in I Ri can be turned into a ring by defining the operations coordinatewise, i.e. (ai) + (bi) = (ai + bi) / (ai) · (bi) = (ai · bi). The resulting ring is called a direct...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, it is possible to combine several rings into one large product ring. This is done as follows: if I is some index set and Ri is a ring for every i in I, then the cartesian product i in I Ri can be turned into a ring by defining the operations coordinatewise, i.e. (ai) + (bi) = (ai + bi) / (ai) · (bi) = (ai · bi). The resulting ring is called a direct product of the rings Ri. The direct product of finitely many rings R1,...,Rk is also written as R1 x R2 x ... x Rk or R1 R2 ... Rk, and can also be called the direct sum (and sometimes the complete direct sum) of the rings Ri. If R = i in I Ri is a product of rings, then for every i in I we have a surjective ring homomorphism pi: R -> Ri which projects the product on the i-th coordinate.
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