Consider the mandatory/optional (participation) nature of the relationship to help determine if it is an identifying relationship.
If the relationship to an entity is not Mandatory - then the relationship to that entity cannot be an identifying relationship.
A 1:1 relationship would generally suggest a flaw in the model under normalisation rules - the data in both entities can be identified by the key in present in either entity. There may be other reasons to split such data into two entities, such as in the case of a very sparsely populated, large and wide data entity - but these tend to be physical considerations rather than logical or conceptual. Although one could model User and User Profile as 1:1 (if there is no need to keep user profile history) - a user profile record requires a user record to exist - it is mandatory - and therefore an identifying relationship. (represented in ER as USER-|----o-PROFILE) - the USER record must be created first so that a Profile record can then be created - so optional in the USER -- PROFILE direction)
A 1:N relationship is often where an identifying relationship is needed - but if the relationship is described as optional then the entity record may or may not store the foreign key identity from that relationship - so it would not be an identifying relationship.
( relationship represented as >|---o- or >|---|- or >o---o- or >o---|-) - excuse my rough crow's feet notation
Performers may be Booked for Performances say. Within the context of the business, the business may be set up such that
either
1. A Performer must have at least one Booking and each Booking must have a performer - mandatory participation in both directions
2. A Performer may have one or more Bookings, however the Booking must involve a performer - mandatory participation is dictated for a Performer in a Booking,
3. A Performer must have at least one Booking, while a Booking may be for one or more Performers - mandatory participation is dictated for Performers however a booking may not involve a performer (eg some other type of booking event - exhibition)
4. A Performer may have one or more Bookings and a Booking may be for one or more Performers - participation is optional in both.
Identifying relationships will only occur where the Booking requires a Performer (Case 1, Case 2). Case 3, 4 do not require a performer to be identified, and the Performer entity does not identify a booking.
For M:N relationships in ER models, the analysis on the basis of participation can be applied to determine in the relationship is identifying - although it appears that if mandatory in both directions you would have the difficulty of there being 2 identifying relationships referring to the other entity - a circular reference and probably not practical. These are ultimately resolved through application of normalization rules. For M:N focus on the relationship in one direction only.
EDIT: Further to the above:
Within EER the concept of Strong and Weak entities are considered:
The relationship must be many-to-one from the weak to the identifying entity. Participation of the weak entity in the relationship must be mandatory. The discriminator (or partial key) of a weak entity distinguishes weak entities that depend on the same specific strong entity. The primary key of a weak entity is the primary key of the identifying entity set + the partial key of the weak entity set.
Eg: Many payments are made on a loan
- Payments don't exist without a loan.
- Multiple loans will each have a first, second payment and so on. So, each payment is only unique in the context of the loan which it is paying off.
The payment is a weak entity; its existence is dependent on the loan entity.
Other cases: SalesOrder- SalesOrderLineItem, Country - State
Generally the logical/conceptual primary key of the weak entity is the composite of the primary key of the strong entity and an identifier for the weak entity within the context of the strong entity. The physical implementation of the weak entity will store the PK of the strong entity, and in many cases, partial key representing the line item (PK of weak entity = PK of strong entity + weak entity partial key). However, in some cases the design may invoke a unique PK for the weak entity, although often there is some means to also represent the order (eg lineorder sequence no) - as a result - these identifying relationships can be somewhat obscured in the physical implementation