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fresh_42

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One more question. Why doesn't a subring necessarily have to have the same multiplicative identity as the bigger ring?

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fresh_42

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Do you have an example? I can only think of examples like ##n\cdot \mathbb{Z} \subseteq \mathbb{Z}## where the subring has none.One more question. Why doesn't a subring necessarily have to have the same multiplicative identity as the bigger ring?

In general, a multiplicative identity doesn't always exist. The multiplicative structure of a ring doesn't need to define a group structure, but if rings are compared like ring and subring and both have a ##1##, then it's usually required to be the same. At least the ##1## in the ring is also a ##1## in the subring, if it's included. If not, and the subring has another element as a ##1## it's getting a bit messy, since this one will act as a ##1## on certain elements of the ring as well - or you have completely different multiplicative structures, in which case one doesn't speak of a subring.

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How about {0,2,4} as a subring of ZDo you have an example? I can only think of examples like n⋅Z⊆Zn⋅Z⊆Zn\cdot \mathbb{Z} \subseteq \mathbb{Z} where the subring has none.

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https://en.wikipedia.org/wiki/Fundamental_theorem_of_linear_algebra

Basically , if A has rows ## a_1,a_2 ,..,a_n ## and B has columns ## b_1, b_2,.., b_n ## then you have that

## a_ i. b_j =0 ## for all ## 0 \leq i,j \leq n ## so that every row of A is orthogonal to every column of B . This implies, by linearity, that the row space of A is orthogonal to the column space of B and the column space of B is contained in the nullspace of A..

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