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In [[ring theory]], an '''ideal''' is defined as a subset of a ring with the following properties:
#Zero is an element of the ideal
#The ideal is closed under addition
#The product of an element of the ideal and an element of the initial ring is an element of the ideal
#Zero is an element of the ideal
#The ideal is closed under addition
#The product of an element of the ideal and an element of the initial ring is an element of the ideal