definition. ring quotient [ca-001B]

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RingQuot

L∃∀N

Let \(R\) be a non-commutative ring, \(r\) an arbitrary equivalence relation on \(R\). The ring quotient of \(R\) by \(r\) is the quotient of \(R\) by the strengthen equivalence relation of \(r\) such that for all \(a, b, c\) in \(R\):

1. \(a + c \sim b + c\) if \(a \sim b\)
2. \(a * c \sim b * c\) if \(a \sim b\)
3. \(a * b \sim a * c\) if \(b \sim c\)