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Note that the sage instances corresponding to the ring $(\Bbb Z/6,+,\cdot)$, and to the cyclic abelian group $(C_6,+)$ differ, as it also happens in the mathematical world. The quotient in the category of rings is built w.r.t. an ideal, in our case the ideal is generated by $3\in \Bbb Z/6$. In the category of abelian groups we mod out a subgroup, in our case the subgroup is generated by $3$. Let us construct the quotient in both cases.

In the category of rings, we introduce the ring $R$ by either of...

sage: Integers(6)                                                                                                                                             
Ring of integers modulo 6
sage: Zmod(6)                                                                                                                                                 
Ring of integers modulo 6
sage: IntegerModRing(6)                                                                                                                                       
Ring of integers modulo 6

then consider its ideal $J=(3)$, and the quotient:

sage: R = Zmod(6)                                                                                                                                             
sage: R                                                                                                                                                       
Ring of integers modulo 6
sage: J = R.ideal(3)                                                                                                                                          
sage: J                                                                                                                                                       
Principal ideal (3) of Ring of integers modulo 6
sage: list(R)                                                                                                                                                 
[0, 1, 2, 3, 4, 5]
sage: Q = R.quotient(J)                                                                                                                                       
sage: Q                                                                                                                                                       
Ring of integers modulo 3
sage: for k in R: 
....:     print(f"{k} modulo J is {Q(k)}") 
0 modulo J is 0
1 modulo J is 1
2 modulo J is 2
3 modulo J is 0
4 modulo J is 1
5 modulo J is 2

In the category of abelian groups (and/or in the category of groups) one can construct:

sage: C6.<a> = AbelianGroup(1, [6])                                                                                                                           
sage: C6                                                                                                                                                      
Multiplicative Abelian group isomorphic to C6
sage: # or                                                                                                                                                    
sage: C6 = CyclicPermutationGroup(6)                                                                                                                          
sage: a = C6.gens()[0]                                                                                                                                        
sage: a.order()                                                                                                                                               
sage: a                                                                                                                                                       

And the quotient is...

sage: H = C6.subgroup([a^3])                                                                                                                                  
sage: Q = C6.quotient(H)                                                                                                                                      
sage: Q                                                                                                                                                       
Permutation Group with generators [(1,2,3)]

Unfortunately, the version using C6.<a> = AbelianGroup(1, [6]) was leading to a sage crash on my machine while trying to build the quotient with respect to the subgroup H constructed mot-a-mot as above. So just use the construction involving a permutation group.